Algebra Quiz: Complete The Square To Find Solutions

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Complete The Square To Find SolutionsQuestion 1 of 20

Solve the quadratic equation $x^2 + 8x + 7 = 0$ by completing the square. Give the solutions for $x$ .

x=-1,\,-7

x=4\pm\sqrt{9}

x=-4\pm\sqrt{9}

x=-4\pm\sqrt{7}

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Algebra Quiz

Algebra Quiz: Complete The Square To Find Solutions

Practice Complete The Square To Find Solutions in Algebra with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Complete The Square To Find Solutions, giving you a quick way to practice the rules, question types, and explanations that matter most for Algebra.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

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Question 1

Solve the quadratic equation $x^2 + 8x + 7 = 0$ by completing the square. Give the solutions for $x$ .

$x=-1,\,-7$ (correct answer)
$x=4\pm\sqrt{9}$
$x=-4\pm\sqrt{9}$
$x=-4\pm\sqrt{7}$

Explanation: This question tests your ability to use completing the square to solve quadratic equations by transforming them into the form (x - p)² = q, which makes finding solutions straightforward by taking square roots. Completing the square is a method that transforms any quadratic equation ax² + bx + c = 0 into an equivalent equation (x - p)² = q that has the same solutions but is much easier to solve: once in this form, we just take the square root of both sides (remembering ±!), giving x - p = ±√q, then solve for x = p ± √q. This method works for ALL quadratics, even ones that don't factor nicely! Solving x² + 8x + 7 = 0 by completing the square: (1) Move constant to right: x² + 8x = -7. (2) Take half of 8 to get 4, square it to get 16, add to both sides: x² + 8x + 16 = -7 + 16. (3) Left side is perfect square (x + 4)², right side simplifies to 9: (x + 4)² = 9. (4) Take square roots: x + 4 = ±√9. (5) Solve: x = -4 ± 3 = -1 or -7. Done! Choice A correctly solves to get x = -1 and x = -7 with accurate arithmetic and proper form. Choice C has the right perfect square form but makes a sign error: from (x + 4)², the p in (x - p)² is -4, not +4. Watch those signs! When we have (x + 4)², we can rewrite as (x - (-4))², so p = -4. The sign in the factored form is opposite to the p value. The complete completing-the-square procedure: (1) If a ≠ 1, divide everything by a first to get x² coefficient = 1, (2) Move the constant term to the right side, (3) Take half the x-coefficient, square it: (b/2)², (4) Add this to BOTH sides, (5) Factor left side as (x + b/2)² (it's a perfect square now!), (6) Simplify right side, (7) Take ± square root of both sides, (8) Solve for x. Follow these 8 steps systematically and you'll get the right answer every time!

Question 2

Transform and solve by completing the square: $x^2 - 6x + 2 = 0$ . First write it in the form $(x-p)^2=q$ , then find the solutions.

$(x-6)^2=2\;\Rightarrow\;x=6\pm\sqrt{2}$
$(x-3)^2=7\;\Rightarrow\;x=3\pm\sqrt{7}$ (correct answer)
$(x+3)^2=7\;\Rightarrow\;x=-3\pm\sqrt{7}$
$(x-3)^2=11\;\Rightarrow\;x=3\pm\sqrt{11}$

Explanation: This question tests your ability to use completing the square to solve quadratic equations by transforming them into the form (x - p)² = q, which makes finding solutions straightforward by taking square roots. The completing-the-square process creates a perfect square trinomial on one side: for x² + bx, we add (b/2)²—half the middle coefficient, squared—to both sides. This turns x² + bx + (b/2)² into (x + b/2)², a perfect square that factors as a binomial squared. Then we solve by taking square roots. It's like setting up a problem in a form where the answer is obvious! Solving x² - 6x + 2 = 0 by completing the square: (1) Move constant to right: x² - 6x = -2. (2) Take half of -6 to get -3, square it to get 9, add to both sides: x² - 6x + 9 = -2 + 9. (3) Left side is perfect square (x - 3)², right side simplifies to 7: (x - 3)² = 7. (4) Take square roots: x - 3 = ±√7. (5) Solve: x = 3 ± √7. Done! Choice A correctly completes the square to get (x - 3)² = 7 and solves to get x = 3 ± √7 with accurate arithmetic and proper form. Choice B has the right perfect square form but makes a sign error: from (x - 3)², the sign inside is negative, but B uses (x + 3)² which would come from a positive coefficient. Watch those signs! When we have (x - 3)², it's equivalent to (x + (-3))², so the sign depends on b. The sign in the factored form is opposite to the p value. The (b/2)² trick: if you have x² + 6x, half of 6 is 3, and 3² = 9, so you add 9 to complete the square, getting (x + 3)². If you have x² - 10x, half of -10 is -5, and (-5)² = 25 (positive!), so add 25 to get (x - 5)². The sign of b affects the sign in the binomial, but (b/2)² is always positive. Practice this on a few examples and it becomes automatic!

Question 3

Derive the quadratic formula by completing the square on the general equation $ax^2 + bx + c = 0$ (with $a\ne 0$ ). Which expression for $x$ results?

$x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$ (correct answer)
$x=\dfrac{-b\pm\sqrt{b^2-2ac}}{a}$
$x=\dfrac{b\pm\sqrt{b^2-4ac}}{2a}$
$x=\dfrac{-b\pm\sqrt{b^2+4ac}}{2a}$

Explanation: This question tests your ability to use completing the square to solve quadratic equations by transforming them into the form (x - p)² = q, which makes finding solutions straightforward by taking square roots. The quadratic formula x = (-b ± √(b² - 4ac))/(2a) actually comes FROM completing the square on the general equation ax² + bx + c = 0! If you complete the square on this general form (treating a, b, c as constants), you derive the quadratic formula. So completing the square isn't just one method among many—it's the fundamental method that gives us the formula itself! Starting with ax² + bx + c = 0: (1) Divide by a: x² + (b/a)x + (c/a) = 0. (2) Move constant: x² + (b/a)x = -c/a. (3) Complete square, half of b/a is b/(2a), squared is b²/(4a²): x² + (b/a)x + b²/(4a²) = -c/a + b²/(4a²). (4) Left side: (x + b/(2a))². Right side: (b² - 4ac)/(4a²). (5) So (x + b/(2a))² = (b² - 4ac)/(4a²). (6) Square root: x + b/(2a) = ±√(b² - 4ac)/(2a). (7) Solve: x = -b/(2a) ± √(b² - 4ac)/(2a) = (-b ± √(b² - 4ac))/(2a). That's the quadratic formula! Choice C correctly derives the formula x = (-b ± √(b² - 4ac))/(2a) with accurate algebra and proper signs throughout the derivation. Choice A has the wrong sign under the radical: when we move c/a to the right and add b²/(4a²), we get (b² - 4ac)/(4a²), not (b² + 4ac)/(4a²). The -4ac term comes from -c/a = -4ac/(4a²) when we find a common denominator. Sign errors in derivations are common but critical! Why this works: when you add (b/2)² to x² + bx, you're creating the expansion of (x + b/2)²: remember (a + b)² = a² + 2ab + b², so (x + b/2)² = x² + 2(x)(b/2) + (b/2)² = x² + bx + (b/2)². That's why adding (b/2)² makes a perfect square! It's not random—it completes the perfect square trinomial pattern.

Question 4

Transform $x^2 - 4x + 1 = 0$ into the form $(x - p)^2 = q$ (an equivalent equation with the same solutions), then solve by taking square roots.

$(x+2)^2=3\;\Rightarrow\; x=-2\pm\sqrt{3}$
$(x-2)^2=3\;\Rightarrow\; x=2\pm\sqrt{3}$ (correct answer)
$(x-2)^2=-3\;\Rightarrow\; x=2\pm\sqrt{3}$
$(x-4)^2=1\;\Rightarrow\; x=4\pm 1$

Explanation: This question tests your ability to use completing the square to solve quadratic equations by transforming them into the form (x - p)² = q, which makes finding solutions straightforward by taking square roots. The completing-the-square process creates a perfect square trinomial on one side: for x² + bx, we add (b/2)²—half the middle coefficient, squared—to both sides. This turns x² + bx + (b/2)² into (x + b/2)², a perfect square that factors as a binomial squared. Then we solve by taking square roots. It's like setting up a problem in a form where the answer is obvious! Solving x² - 4x + 1 = 0 by completing the square: (1) Move constant to right: x² - 4x = -1. (2) Take half of -4 to get -2, square it to get 4, add to both sides: x² - 4x + 4 = -1 + 4. (3) Left side is perfect square (x - 2)², right side simplifies to 3: (x - 2)² = 3. (4) Take square roots: x - 2 = ±√3. (5) Solve: x = 2 ± √3. Done! Choice B correctly completes the square to get (x - 2)² = 3 and solves to get x = 2 ± √3 with accurate arithmetic and proper form. Choice A has a sign error: from x² - 4x, half of -4 is -2, so we get (x - 2)², not (x + 2)². Watch those signs! When we have x² - 4x, the middle term is negative, so we get (x - 2)², not (x + 2)². The sign in the factored form matches the sign of half the middle coefficient. The (b/2)² trick: if you have x² + 6x, half of 6 is 3, and 3² = 9, so you add 9 to complete the square, getting (x + 3)². If you have x² - 10x, half of -10 is -5, and (-5)² = 25 (positive!), so add 25 to get (x - 5)². The sign of b affects the sign in the binomial, but (b/2)² is always positive. Practice this on a few examples and it becomes automatic!

Question 5

Show that $x^2 + 4x - 5 = 0$ and $(x+2)^2 = 9$ have the same solutions by solving the completed-square equation and listing the solution set.

$x=1$ or $x=-5$ (correct answer)
$x=-2$ only
$x=-1$ or $x=5$
$x=2$ or $x=-2$

Explanation: This question tests your ability to use completing the square to solve quadratic equations by transforming them into the form (x - p)² = q, which makes finding solutions straightforward by taking square roots. The completing-the-square process creates a perfect square trinomial on one side: for x² + bx, we add (b/2)²—half the middle coefficient, squared—to both sides. This turns x² + bx + (b/2)² into (x + b/2)², a perfect square that factors as a binomial squared. Then we solve by taking square roots. It's like setting up a problem in a form where the answer is obvious! Solving (x + 2)² = 9 by taking square roots: (1) Take square roots of both sides: x + 2 = ±3. (2) Solve for x: x = -2 ± 3. (3) This gives x = -2 + 3 = 1 or x = -2 - 3 = -5. To verify these are the same solutions as x² + 4x - 5 = 0, we can complete the square on that equation: x² + 4x = 5, add 4 to both sides to get x² + 4x + 4 = 9, which factors as (x + 2)² = 9—the same equation! Choice B correctly solves (x + 2)² = 9 to get x = -2 ± 3, which gives x = 1 or x = -5, with accurate arithmetic and proper form. Choice C reverses the signs: from x + 2 = ±3, we get x = -2 ± 3, not x = 2 ± 3. When solving x + 2 = 3, we subtract 2 to get x = 1, not add 2 to get x = 5. Be careful with the algebra when isolating x! To verify your completing-the-square work: expand your (x - p)² + k form back out using FOIL, and you should get back to your original quadratic. If you don't, there's an error. Also, solve the original equation using the quadratic formula—you should get the same solutions. These double-checks catch mistakes and build confidence!

Question 6

What is the completed square form of $x^2+4x-5=0$ written as $(x-p)^2=q$ ?

$(x+2)^2=1$
$(x+2)^2=9$ (correct answer)
$(x-2)^2=-9$
$(x-2)^2=9$

Explanation: This question tests your ability to use completing the square to solve quadratic equations by transforming them into the form (x - p)² = q, which makes finding solutions straightforward by taking square roots. The form (x - p)² = q tells us immediately: if q > 0, there are two real solutions (x = p ± √q); if q = 0, there's one solution (x = p); if q < 0, there are no real solutions (we'd need to take square root of negative). So completing the square not only solves the equation, it also reveals the nature of the solutions before we calculate them! Solving x² + 4x - 5 = 0 by completing the square: (1) Move constant to right: x² + 4x = 5. (2) Take half of 4 to get 2, square it to get 4, add to both sides: x² + 4x + 4 = 5 + 4. (3) Left side is perfect square (x + 2)², right side simplifies to 9: (x + 2)² = 9. (4) Take square roots: x + 2 = ±3. (5) Solve: x = -2 ± 3. Done! Choice C correctly completes the square to get (x + 2)² = 9 with accurate arithmetic and proper form. Choice D doesn't add (b/2)² to both sides—only adds to one side or miscalculates, leading to negative q incorrectly! Completing the square requires adding the perfect square term to BOTH sides to maintain equality. If you only add to the left, you've changed the equation and will get wrong solutions. Both sides, always! The ± is crucial: from (x - 3)² = 16, taking square roots gives x - 3 = ±4 (both +4 and -4), so x = 3 + 4 = 7 OR x = 3 - 4 = -1. Two solutions! Don't forget the ± and don't forget to split it into two separate solutions. Check both: 7 and -1 both satisfy the original equation? Yes!

Question 7

Derive the quadratic formula by completing the square on the general equation $ax^2+bx+c=0$ (assume $a\ne0$ ). Which expression correctly gives the solutions for $x$ ?

$x=\dfrac{-b\pm\sqrt{b^2+4ac}}{2a}$
$x=\dfrac{b\pm\sqrt{b^2-4ac}}{2a}$
$x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$ (correct answer)
$x=\dfrac{-b\pm\sqrt{b^2-2ac}}{a}$

Explanation: This question tests your ability to use completing the square to solve quadratic equations by transforming them into the form (x - p)² = q, which makes finding solutions straightforward by taking square roots. The quadratic formula x = (-b ± √(b² - 4ac))/(2a) actually comes FROM completing the square on the general equation ax² + bx + c = 0! If you complete the square on this general form (treating a, b, c as constants), you derive the quadratic formula. So completing the square isn't just one method among many—it's the fundamental method that gives us the formula itself! Starting with ax² + bx + c = 0: (1) Divide by a: x² + (b/a)x + (c/a) = 0. (2) Move constant: x² + (b/a)x = -c/a. (3) Complete square, half of b/a is b/(2a), squared is b²/(4a²): x² + (b/a)x + b²/(4a²) = -c/a + b²/(4a²). (4) Left side: (x + b/(2a))². Right side: (b² - 4ac)/(4a²). (5) So (x + b/(2a))² = (b² - 4ac)/(4a²). (6) Square root: x + b/(2a) = ±√(b² - 4ac)/(2a). (7) Solve: x = -b/(2a) ± √(b² - 4ac)/(2a) = (-b ± √(b² - 4ac))/(2a). That's the quadratic formula! Choice C correctly derives the formula x = (-b ± √(b² - 4ac))/(2a) with accurate arithmetic and proper form. Choice A has the wrong sign under the radical: it should be b² - 4ac, not b² + 4ac. The discriminant b² - 4ac comes from combining the fractions on the right side when completing the square. This sign is crucial for determining the nature of solutions! Why this works: when you add (b/2)² to x² + bx, you're creating the expansion of (x + b/2)²: remember (a + b)² = a² + 2ab + b², so (x + b/2)² = x² + 2(x)(b/2) + (b/2)² = x² + bx + (b/2)². That's why adding (b/2)² makes a perfect square! It's not random—it completes the perfect square trinomial pattern.

Question 8

Transform and solve: $x^2 - 6x + 2 = 0$ . Use completing the square to write it as $(x - p)^2 = q$ and then find $x$ .

$(x+3)^2=7\;\Rightarrow\; x=-3\pm\sqrt{7}$
$(x-3)^2=7\;\Rightarrow\; x=3\pm\sqrt{7}$ (correct answer)
$(x-6)^2=2\;\Rightarrow\; x=6\pm\sqrt{2}$
$(x-3)^2=11\;\Rightarrow\; x=3\pm\sqrt{11}$

Explanation: This question tests your ability to use completing the square to solve quadratic equations by transforming them into the form (x - p)² = q, which makes finding solutions straightforward by taking square roots. Completing the square is a method that transforms any quadratic equation ax² + bx + c = 0 into an equivalent equation (x - p)² = q that has the same solutions but is much easier to solve: once in this form, we just take the square root of both sides (remembering ±!), giving x - p = ±√q, then solve for x = p ± √q. This method works for ALL quadratics, even ones that don't factor nicely! Solving x² - 6x + 2 = 0 by completing the square: (1) Move constant to right: x² - 6x = -2. (2) Take half of -6 to get -3, square it to get 9, add to both sides: x² - 6x + 9 = -2 + 9. (3) Left side is perfect square (x - 3)², right side simplifies to 7: (x - 3)² = 7. (4) Take square roots: x - 3 = ±√7. (5) Solve: x = 3 ± √7. Done! Choice A correctly completes the square to get (x - 3)² = 7 and solves to get x = 3 ± √7 with accurate arithmetic and proper form. Choice C makes an arithmetic error when simplifying the right side: -2 + 9 = 7, not 11, so we should get (x - 3)² = 7, not (x - 3)² = 11. Always double-check your arithmetic when combining terms on the right side after adding (b/2)² to both sides! Why this works: when you add (b/2)² to x² + bx, you're creating the expansion of (x + b/2)²: remember (a + b)² = a² + 2ab + b², so (x + b/2)² = x² + 2(x)(b/2) + (b/2)² = x² + bx + (b/2)². That's why adding (b/2)² makes a perfect square! It's not random—it completes the perfect square trinomial pattern.

Question 9

Derive the quadratic formula by completing the square on the general equation $ax^2+bx+c=0$ (assume $a\ne 0$ ). Which expression for $x$ results?

$x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$ (correct answer)
$x=\dfrac{b\pm\sqrt{b^2-4ac}}{2a}$
$x=\dfrac{-b\pm\sqrt{b^2+4ac}}{2a}$
$x=\dfrac{-b\pm\sqrt{(b-2a)^2-4ac}}{a}$

Explanation: This question tests your ability to use completing the square to solve quadratic equations by transforming them into the form (x - p)² = q, which makes finding solutions straightforward by taking square roots. The quadratic formula x = (-b ± √(b² - 4ac))/(2a) actually comes FROM completing the square on the general equation ax² + bx + c = 0! If you complete the square on this general form (treating a, b, c as constants), you derive the quadratic formula. So completing the square isn't just one method among many—it's the fundamental method that gives us the formula itself! Starting with ax² + bx + c = 0: (1) Divide by a: x² + (b/a)x + (c/a) = 0. (2) Move constant: x² + (b/a)x = -c/a. (3) Complete square, half of b/a is b/(2a), squared is b²/(4a²): x² + (b/a)x + b²/(4a²) = -c/a + b²/(4a²). (4) Left side: (x + b/(2a))². Right side: (b² - 4ac)/(4a²). (5) So (x + b/(2a))² = (b² - 4ac)/(4a²). (6) Square root: x + b/(2a) = ±√(b² - 4ac)/(2a). (7) Solve: x = -b/(2a) ± √(b² - 4ac)/(2a) = (-b ± √(b² - 4ac))/(2a). That's the quadratic formula! Choice A correctly derives the formula with accurate arithmetic and proper form. Choice B makes a sign error when solving: the term is -b/(2a), not +b/(2a), because we subtract b/(2a) after the square root step. Completing the square involves several arithmetic steps, especially when deriving the formula with a, b, c. Take your time and double-check fraction arithmetic! Why this works: when you add (b/2)² to x² + bx, you're creating the expansion of (x + b/2)²: remember (a + b)² = a² + 2ab + b², so (x + b/2)² = x² + 2(x)(b/2) + (b/2)² = x² + bx + (b/2)². That's why adding (b/2)² makes a perfect square! It's not random—it completes the perfect square trinomial pattern.

Question 10

Derive the quadratic formula by completing the square on the general equation $ax^2+bx+c=0$ (assume $a\ne0$ ). Which expression for $x$ results?

$x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$ (correct answer)
$x=\dfrac{-b\pm\sqrt{b^2-2ac}}{a}$
$x=\dfrac{b\pm\sqrt{b^2-4ac}}{2a}$
$x=\dfrac{-b\pm\sqrt{b^2+4ac}}{2a}$

Explanation: This question tests your ability to use completing the square to solve quadratic equations by transforming them into the form (x - p)² = q, which makes finding solutions straightforward by taking square roots. The quadratic formula x = (-b ± √(b² - 4ac))/(2a) actually comes FROM completing the square on the general equation ax² + bx + c = 0! If you complete the square on this general form (treating a, b, c as constants), you derive the quadratic formula. So completing the square isn't just one method among many—it's the fundamental method that gives us the formula itself! Starting with ax² + bx + c = 0: (1) Divide by a: x² + (b/a)x + (c/a) = 0. (2) Move constant: x² + (b/a)x = -c/a. (3) Complete square, half of b/a is b/(2a), squared is b²/(4a²): x² + (b/a)x + b²/(4a²) = -c/a + b²/(4a²). (4) Left side: (x + b/(2a))². Right side: (b² - 4ac)/(4a²). (5) So (x + b/(2a))² = (b² - 4ac)/(4a²). (6) Square root: x + b/(2a) = ±√(b² - 4ac)/(2a). (7) Solve: x = -b/(2a) ± √(b² - 4ac)/(2a) = (-b ± √(b² - 4ac))/(2a). That's the quadratic formula! Choice C correctly derives the formula with accurate arithmetic and proper form. Choice A makes an arithmetic error when combining fractions in derivation: the right side is (b² - 4ac)/(4a²), not +4ac; completing the square involves several arithmetic steps, especially when deriving the formula with a, b, c. Take your time and double-check fraction arithmetic! To verify your completing-the-square work: expand your (x - p)² + k form back out using FOIL, and you should get back to your original quadratic. If you don't, there's an error. Also, solve the original equation using the quadratic formula—you should get the same solutions. These double-checks catch mistakes and build confidence!

Question 11

Complete the square to solve $x^2+6x+5=0$ . Write an equivalent equation in the form $(x-p)^2=q$ , then find the solutions.

$(x+3)^2=14$ , so $x=-3\pm\sqrt{14}$
$(x+3)^2=4$ , so $x=-3\pm2$ (correct answer)
$(x+6)^2=31$ , so $x=-6\pm\sqrt{31}$
$(x-3)^2=4$ , so $x=3\pm2$

Explanation: This question tests your ability to use completing the square to solve quadratic equations by transforming them into the form (x - p)² = q, which makes finding solutions straightforward by taking square roots. The completing-the-square process creates a perfect square trinomial on one side: for x² + bx, we add (b/2)²—half the middle coefficient, squared—to both sides. This turns x² + bx + (b/2)² into (x + b/2)², a perfect square that factors as a binomial squared. Then we solve by taking square roots. It's like setting up a problem in a form where the answer is obvious! Solving x² + 6x + 5 = 0 by completing the square: (1) Move constant to right: x² + 6x = -5. (2) Take half of 6 to get 3, square it to get 9, add to both sides: x² + 6x + 9 = -5 + 9. (3) Left side is perfect square (x + 3)², right side simplifies to 4: (x + 3)² = 4. (4) Take square roots: x + 3 = ±2. (5) Solve: x = -3 ± 2, so x = -1 or x = -5. Done! Choice B correctly completes the square to get (x + 3)² = 4 and solves to get x = -3 ± 2 with accurate arithmetic and proper form. Choice A makes an arithmetic error when simplifying the right side: -5 + 9 = 4, not 14. Completing the square involves several arithmetic steps. Take your time and double-check fraction arithmetic! Why this works: when you add (b/2)² to x² + bx, you're creating the expansion of (x + b/2)²: remember (a + b)² = a² + 2ab + b², so (x + b/2)² = x² + 2(x)(b/2) + (b/2)² = x² + bx + (b/2)². That's why adding (b/2)² makes a perfect square! It's not random—it completes the perfect square trinomial pattern.

Question 12

Complete the square to solve $2x^2+12x+10=0$ . (Do not use factoring or the quadratic formula.)

$x=-3\pm\sqrt2$
$x=-3\pm2$ (correct answer)
$x=-6\pm\sqrt2$
$x=3\pm\sqrt2$

Explanation: This question tests your ability to use completing the square to solve quadratic equations by transforming them into the form (x - p)² = q, which makes finding solutions straightforward by taking square roots. Completing the square is a method that transforms any quadratic equation ax² + bx + c = 0 into an equivalent equation (x - p)² = q that has the same solutions but is much easier to solve: once in this form, we just take the square root of both sides (remembering ±!), giving x - p = ±√q, then solve for x = p ± √q. This method works for ALL quadratics, even ones that don't factor nicely! When a ≠ 1, like 2x² + 12x + 10 = 0: (1) First divide everything by 2: x² + 6x + 5 = 0. (2) Now complete the square on this simpler equation: x² + 6x = -5, add (6/2)² = 9 to both sides: x² + 6x + 9 = -5 + 9 = 4. (3) Factor left: (x + 3)² = 4. (4) Solve: x = -3 ± 2, so x = -1 or x = -5. Dividing by a first makes the completing square much cleaner! Choice B correctly completes the square to get (x + 3)² = 4 and solves to get x = -3 ± 2 with accurate arithmetic and proper form. Choice A calculates (b/2)² incorrectly: half of 6 is 3, and squaring gives 9, not something leading to ±√2; this perfect square term is the heart of completing the square—getting it right is crucial! An error here throws off everything that follows. Why this works: when you add (b/2)² to x² + bx, you're creating the expansion of (x + b/2)²: remember (a + b)² = a² + 2ab + b², so (x + b/2)² = x² + 2(x)(b/2) + (b/2)² = x² + bx + (b/2)². That's why adding (b/2)² makes a perfect square! It's not random—it completes the perfect square trinomial pattern.

Question 13

Solve $2x^2 + 12x + 10 = 0$ by completing the square (divide first if needed).

$x=-3\pm\sqrt{2}$
$x=-1$ or $x=-5$ (correct answer)
$x=3\pm\sqrt{2}$
$x=-3\pm 2$

Explanation: This question tests your ability to use completing the square to solve quadratic equations by transforming them into the form (x - p)² = q, which makes finding solutions straightforward by taking square roots. Completing the square is a method that transforms any quadratic equation ax² + bx + c = 0 into an equivalent equation (x - p)² = q that has the same solutions but is much easier to solve: once in this form, we just take the square root of both sides (remembering ±!), giving x - p = ±√q, then solve for x = p ± √q. This method works for ALL quadratics, even ones that don't factor nicely! When a ≠ 1, like 2x² + 12x + 10 = 0: (1) First divide everything by 2: x² + 6x + 5 = 0. (2) Now complete the square on this simpler equation: x² + 6x = -5, add (6/2)² = 9 to both sides: x² + 6x + 9 = -5 + 9 = 4. (3) Factor left: (x + 3)² = 4. (4) Solve: x + 3 = ±2, so x = -3 ± 2, giving x = -1 or x = -5. Dividing by a first makes the completing square much cleaner! Choice C correctly solves to get x = -1 or x = -5 with accurate arithmetic and proper form. Choice A forgets to simplify after taking the square root: from (x + 3)² = 4, we get x + 3 = ±2 (not ±√4), since √4 = 2. Always simplify radicals: √4 = 2, √9 = 3, √16 = 4, etc. Don't leave them as √4 when they simplify to whole numbers! To verify your completing-the-square work: expand your (x - p)² + k form back out using FOIL, and you should get back to your original quadratic. If you don't, there's an error. Also, solve the original equation using the quadratic formula—you should get the same solutions. These double-checks catch mistakes and build confidence!

Question 14

A quadratic equation is given as $x^2 - 6x + 2 = 0$ . When completing the square to transform this equation into the form $(x - p)^2 = q$ , what is the value of $q$ ?

$7$ (correct answer)
$9$
$-7$
$11$

Explanation: To complete the square for

x^2 - 6x + 2 = 0

, first move the constant:

x^2 - 6x = -2

. Take half of the coefficient of

x

and square it:

(-6/2)^2 = 9

. Add 9 to both sides:

x^2 - 6x + 9 = -2 + 9 = 7

. This gives

(x - 3)^2 = 7

, so

q = 7

. Choice B incorrectly uses the value added to complete the square. Choice C uses

-7

from a sign error. Choice D results from adding instead of subtracting the constant term.

Question 15

The equation $x^2 + kx + 12 = 0$ is completed to the form $(x + \frac{k}{2})^2 = q$ . If this quadratic equation has exactly one solution, what is the value of $k$ and what is that solution?

$k = 0$ ; solution is $x = \pm 2\sqrt{3}$
$k = \pm 4\sqrt{3}$ ; solution is $x = \mp 2\sqrt{3}$ (correct answer)
$k = \pm 4\sqrt{3}$ ; solution is $x = 0$
$k = 12$ ; solution is $x = -6$

Explanation: When you see a quadratic that "has exactly one solution," this means the discriminant equals zero, creating a perfect square trinomial. The key insight is understanding what happens when you complete the square and how this connects to the number of solutions. Starting with

x^2 + kx + 12 = 0

, let's complete the square. Moving the constant term:

x^2 + kx = -12

. Adding

(\frac{k}{2})^2

to both sides:

(x + \frac{k}{2})^2 = -12 + \frac{k^2}{4}

. This gives us

q = \frac{k^2}{4} - 12

. For exactly one solution, the right side must equal zero:

\frac{k^2}{4} - 12 = 0

. Solving:

\frac{k^2}{4} = 12

, so

k^2 = 48

, which means

k = \pm 4\sqrt{3}

. When

q = 0

, we have

(x + \frac{k}{2})^2 = 0

, giving us

x = -\frac{k}{2}

. Since

k = \pm 4\sqrt{3}

, the solution is

x = \mp 2\sqrt{3}

(note the sign change). Choice A incorrectly assumes

k = 0

, which would give

q = -12

, resulting in no real solutions. Choice C correctly finds

k = \pm 4\sqrt{3}

but claims the solution is

x = 0

, which ignores that

x = -\frac{k}{2} \neq 0

. Choice D uses

k = 12

, but this gives

q = 24

, resulting in two solutions, not one. Remember: "exactly one solution" means the discriminant is zero, which creates a perfect square. Always check that your final answer produces the required number of solutions.

Question 16

A student is completing the square for $x^2 + 5x - 3 = 0$ and writes the intermediate step: $x^2 + 5x + \frac{25}{4} = 3 + \frac{25}{4}$ . What should be the next step in their work?

Write $(x + \frac{5}{2})^2 = \frac{37}{4}$ and solve for $x$ (correct answer)
Write $(x - \frac{5}{2})^2 = \frac{37}{4}$ and solve for $x$
Simplify the right side to get $\frac{12}{4} + \frac{25}{4} = \frac{37}{4}$
Write $(x + \frac{5}{2})^2 = \frac{12}{4} + \frac{25}{4}$ without simplifying

Explanation: The student correctly added

\frac{25}{4}

to both sides. The left side factors as

(x + \frac{5}{2})^2

since we added

(\frac{5}{2})^2 = \frac{25}{4}

. The right side simplifies to

3 + \frac{25}{4} = \frac{12}{4} + \frac{25}{4} = \frac{37}{4}

. The next step is to write the factored form

(x + \frac{5}{2})^2 = \frac{37}{4}

. Choice B has the wrong sign. Choice C focuses on arithmetic rather than the algebraic transformation. Choice D leaves the work incomplete.

Question 17

A teacher asks students to derive the quadratic formula by completing the square on $ax^2 + bx + c = 0$ . After dividing by $a$ and completing the square, a student obtains $\left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}$ . What is the next step to arrive at the quadratic formula?

Take the square root of both sides to get $x + \frac{b}{2a} = \pm\frac{\sqrt{b^2 - 4ac}}{2|a|}$
Take the square root of both sides to get $x + \frac{b}{2a} = \pm\frac{\sqrt{b^2 - 4ac}}{2a}$ (correct answer)
Multiply both sides by $4a^2$ to eliminate the denominator on the right side
Expand the left side and rearrange to standard form before taking the square root

Explanation: When you're completing the square to derive the quadratic formula, you're working toward isolating the variable x. At this stage, you have a perfect square on the left side equal to a fraction on the right side, so the logical next step is to eliminate the square by taking the square root of both sides. Taking the square root of both sides gives you

x + \frac{b}{2a} = \pm\frac{\sqrt{b^2 - 4ac}}{2a}

, which is answer choice B. When you take the square root of a fraction, you take the square root of both the numerator and denominator separately. The square root of

4a^2

2|a|

, but since we're in the context of deriving a general formula, we write this as

2a

with the understanding that the

\pm

symbol will account for both positive and negative cases. Choice A incorrectly includes absolute value bars around the

a

in the denominator. While

\sqrt{4a^2} = 2|a|

is technically correct, in the context of the quadratic formula derivation, we handle the sign through the

\pm

symbol rather than absolute values. Choice C suggests multiplying by

4a^2

, which would eliminate the fraction but move you away from isolating x—the opposite of what you want at this step. Choice D suggests expanding and rearranging, which would completely undo the completing the square work you just did. Remember: when solving equations with perfect squares, always take the square root of both sides as your next move, and don't forget the

\pm

symbol since squares have two possible roots.

Question 18

A quadratic equation $ax^2 + bx + c = 0$ with $a \neq 1$ is being solved by completing the square. After the first step of making the leading coefficient 1, the equation becomes $x^2 - \frac{3}{2}x + \frac{1}{4} = 0$ . What was the original equation if all original coefficients were integers?

$8x^2 - 12x + 2 = 0$
$2x^2 - 3x + \frac{1}{2} = 0$
$4x^2 - 6x + 1 = 0$ (correct answer)
$4x^2 + 6x + 1 = 0$

Explanation: When you encounter a quadratic equation problem involving completing the square, remember that the first step is dividing the entire equation by the leading coefficient to make it equal to 1. Since the transformed equation is

x^2 - \frac{3}{2}x + \frac{1}{4} = 0

, you need to work backwards to find the original equation. If the original equation was

ax^2 + bx + c = 0

, then dividing by

a

gives you

x^2 + \frac{b}{a}x + \frac{c}{a} = 0

. This means

\frac{b}{a} = -\frac{3}{2}

and

\frac{c}{a} = \frac{1}{4}

. Since all original coefficients were integers, you need to find integer values where these ratios work out. Testing the answer choices by dividing each by their leading coefficient: For choice C,

4x^2 - 6x + 1 = 0

divided by 4 gives

x^2 - \frac{6}{4}x + \frac{1}{4} = 0

, which simplifies to

x^2 - \frac{3}{2}x + \frac{1}{4} = 0

. This matches perfectly. Choice A gives

x^2 - \frac{3}{2}x + \frac{1}{4} = 0

when divided by 8, but this matches our target equation, making A seem correct too. However, choice A has a leading coefficient of 8 while choice C has 4 - the problem asks for the original equation, and choice C uses smaller integer coefficients, making it the most reduced form. Choice B contains a non-integer coefficient

\frac{1}{2}

, violating the "all integers" condition. Choice D has the wrong sign on the middle term. Strategy tip: When working backwards from a simplified equation, always check that your original coefficients are integers and in their simplest whole-number form.

Question 19

What is the completed-square form of $x^2 + 6x + 5 = 0$ (an equivalent equation with the same solutions) written as $(x-p)^2=q$ ?

$(x+3)^2=4$ (correct answer)
$(x-3)^2=4$
$(x+6)^2=31$
$(x+3)^2=-4$

Explanation: This question tests your ability to use completing the square to solve quadratic equations by transforming them into the form (x - p)² = q, which makes finding solutions straightforward by taking square roots. The completing-the-square process creates a perfect square trinomial on one side: for x² + bx, we add (b/2)²—half the middle coefficient, squared—to both sides. This turns x² + bx + (b/2)² into (x + b/2)², a perfect square that factors as a binomial squared. Then we solve by taking square roots. It's like setting up a problem in a form where the answer is obvious! Solving x² + 6x + 5 = 0 by completing the square: (1) Move constant to right: x² + 6x = -5. (2) Take half of 6 to get 3, square it to get 9, add to both sides: x² + 6x + 9 = -5 + 9. (3) Left side is perfect square (x + 3)², right side simplifies to 4: (x + 3)² = 4. (4) Take square roots: x + 3 = ±2. (5) Solve: x = -3 ± 2. Done! Choice A correctly completes the square to get (x + 3)² = 4 with accurate arithmetic and proper form. Choice B has the right perfect square form but makes a sign error: from (x + 3)², the p in (x - p)² is -3, not +3. Watch those signs! When we have (x + 3)², we can rewrite as (x - (-3))², so p = -3. The sign in the factored form is opposite to the p value. Why this works: when you add (b/2)² to x² + bx, you're creating the expansion of (x + b/2)²: remember (a + b)² = a² + 2ab + b², so (x + b/2)² = x² + 2(x)(b/2) + (b/2)² = x² + bx + (b/2)². That's why adding (b/2)² makes a perfect square! It's not random—it completes the perfect square trinomial pattern.

Question 20

Solve $x^2+2x+5=0$ by completing the square. What does the completed square form reveal about the solutions?

$(x+1)^2=4$ , so $x=-1\pm2$
$(x+1)^2=-4$ , so there are no real solutions (correct answer)
$(x-1)^2=-4$ , so there are no real solutions
$(x+2)^2=1$ , so $x=-2\pm1$

Explanation: This question tests your ability to use completing the square to solve quadratic equations by transforming them into the form (x - p)² = q, which makes finding solutions straightforward by taking square roots. The quadratic formula x = (-b ± √(b² - 4ac))/(2a) actually comes FROM completing the square on the general equation ax² + bx + c = 0! If you complete the square on this general form (treating a, b, c as constants), you derive the quadratic formula. So completing the square isn't just one method among many—it's the fundamental method that gives us the formula itself! After completing the square to get (x - p)² = q, we can see: if q > 0, since q is positive, we can take its square root, giving two real solutions x = p ± √q; if q = 0, we have a perfect square equal to zero, giving one solution x = p; if q < 0, we'd need √(negative), so no real solutions exist—we'd get complex solutions x = p ± i√|q|. The value of q immediately tells us how many real solutions exist! Choice A correctly completes the square to get (x + 1)² = -4, revealing no real solutions with accurate arithmetic and proper form. Choice B makes an arithmetic error when simplifying the right side: -5 + 1 = -4, not +4; completing the square involves several arithmetic steps, so take your time and double-check! To verify your completing-the-square work: expand your (x - p)² + k form back out using FOIL, and you should get back to your original quadratic. If you don't, there's an error. Also, solve the original equation using the quadratic formula—you should get the same solutions. These double-checks catch mistakes and build confidence!