Algebra 2 Quiz: Solve Quadratics By Multiple Methods
20 questions · exam conditions
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Solve Quadratics By Multiple MethodsQuestion 1 of 20

Solve by completing the square: x2+2x+5=0x^2+2x+5=0. (Give solutions in a±bia\pm bi form.)

x=−1±2ix=-1\pm 2i
x=1±2ix=1\pm 2i
x=−1±4ix=-1\pm 4i
x=−2±ix=-2\pm i
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Algebra 2 Quiz

Algebra 2 Quiz: Solve Quadratics By Multiple Methods

Practice Solve Quadratics By Multiple Methods in Algebra 2 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 Solve Quadratics By Multiple Methods, giving you a quick way to practice the rules, question types, and explanations that matter most for Algebra 2.

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.

All questions

Question 1

Solve by completing the square: x2+2x+5=0x^2+2x+5=0. (Give solutions in a±bia\pm bi form.)

  1. x=−1±2ix=-1\pm 2i (correct answer)
  2. x=1±2ix=1\pm 2i
  3. x=−1±4ix=-1\pm 4i
  4. x=−2±ix=-2\pm i
Explanation: This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's x squared = number, inspect or take square roots (x = plus or minus square root of number), (2) if it's (expression) squared = number, take square roots, (3) if it factors easily (like x squared + 5x + 6), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula x = (-b plus or minus square root of (b squared - 4ac)) divided by (2a)—it always works!, (5) if asked to derive the formula, complete the square on general form. Matching method to form saves time and reduces errors! For solving x² + 2x + 5 = 0 by completing the square: move the constant to the right side to get x² + 2x = -5. To complete the square, take half of the x-coefficient (which is 1), square it (getting 1), and add to both sides: x² + 2x + 1 = -5 + 1 = -4. This gives (x + 1)² = -4. Taking square roots: x + 1 = ±2i, so x = -1 ± 2i. Choice A correctly completes the square to get (x + 1)² = -4, then takes square roots to find x = -1 ± 2i, properly handling the negative value under the square root. Choice B incorrectly has +1 as the real part instead of -1, perhaps solving x - 1 = ±2i instead of x + 1 = ±2i—remember when (x + 1)² = -4, we get x + 1 = ±2i, so x = -1 ± 2i! Complex solution recognition: When completing the square yields (expression)² = negative number, you'll get complex solutions. For x² + 2x + 5 = 0: complete to (x + 1)² = -4, so x + 1 = ±2i, giving x = -1 ± 2i. The negative on the right side of (x + 1)² = -4 signals complex solutions!

Question 2

Solve and express solutions in a±bia \pm bi form: x2+4x+8=0x^2 + 4x + 8 = 0.

  1. x=−4±2ix = -4 \pm 2i
  2. x=−2±ix = -2 \pm i
  3. x=2±2ix = 2 \pm 2i
  4. x=−2±2ix = -2 \pm 2i (correct answer)
Explanation: This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's x squared = number, inspect or take square roots (x = plus or minus square root of number), (2) if it's (expression) squared = number, take square roots, (3) if it factors easily (like x squared + 5x + 6), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula x = (-b plus or minus square root of (b squared - 4ac)) divided by (2a)—it always works!, (5) if asked to derive the formula, complete the square on general form. For solving x squared + 4x + 8 = 0: identify a = 1, b = 4, c = 8. Calculate discriminant: b squared - 4ac = 16 - 32 = -16 (negative, so complex solutions!). Apply formula: x = (-4 plus or minus square root of -16) divided by 2 = (-4 plus or minus 4i) divided by 2 = -2 plus or minus 2i. The negative discriminant signals that i will appear in the answer. Choice A correctly identifies x = -2 plus or minus 2i as the complex solutions in a plus or minus bi form. Choice B incorrectly has x = 2 plus or minus 2i, forgetting that -b/2a = -4/2 = -2, not 2. When b = 4, we get -b = -4 in the numerator, and dividing by 2a = 2 gives -2 for the real part. Complex solution recognition: for x squared + 4x + 8 = 0, discriminant = 16 - 32 = -16 (negative!), so complex solutions: x = (-4 plus or minus square root of -16) divided by 2 = (-4 plus or minus 4i) divided by 2 = -2 plus or minus 2i. The pattern is clear: real part = -b/(2a) = -4/2 = -2, imaginary part = plus or minus square root of |discriminant|/(2a) = plus or minus 4/2 = plus or minus 2.

Question 3

Use the quadratic formula to solve: 2x2−3x−2=02x^2 - 3x - 2 = 0.​

  1. x=−3±254x=\dfrac{-3\pm\sqrt{25}}{4}
  2. x=3±174x=\dfrac{3\pm\sqrt{17}}{4}
  3. x=3±52x=\dfrac{3\pm 5}{2}
  4. x=3±254x=\dfrac{3\pm\sqrt{25}}{4} (correct answer)
Explanation: This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's x squared = number, inspect or take square roots (x = plus or minus square root of number), (2) if it's (expression) squared = number, take square roots, (3) if it factors easily (like x squared + 5x + 6), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula x = (-b plus or minus square root of (b squared - 4ac)) divided by (2a)—it always works!, (5) if asked to derive the formula, complete the square on general form. Matching method to form saves time and reduces errors! For 2x squared - 3x - 2 = 0, a=2, b=-3, c=-2; discriminant=9+16=25; x=[3 ± 5]/4, so x=2 or x=-0.5. The formula handles the leading coefficient easily. Choice A correctly applies the formula with -b as positive 3 and proper denominator 4. Choice B forgets the sign of -b, flipping it to negative—double-check signs! Formula tip: write it out fully before plugging in values to avoid errors. Practice on quadratics with a≠1 to get comfortable.

Question 4

Which equation is most efficient to solve by factoring (rather than the quadratic formula)?

  1. 3x2−5x+1=03x^2 - 5x + 1 = 0
  2. 2x2+x−7=02x^2 + x - 7 = 0
  3. x2+2x+5=0x^2 + 2x + 5 = 0
  4. x2+5x+6=0x^2 + 5x + 6 = 0 (correct answer)
Explanation: This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's x squared = number, inspect or take square roots (x = plus or minus square root of number), (2) if it's (expression) squared = number, take square roots, (3) if it factors easily (like x squared + 5x + 6), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula x = (-b plus or minus square root of (b squared - 4ac)) divided by (2a)—it always works!, (5) if asked to derive the formula, complete the square on general form. Let's check which equation factors easily: (A) x squared + 2x + 5: need two numbers that multiply to 5 and add to 2—only 1 and 5, but they add to 6, not 2. Doesn't factor over integers. (B) 3x squared - 5x + 1: with leading coefficient 3, factoring is tricky. (C) x squared + 5x + 6: need two numbers that multiply to 6 and add to 5—that's 2 and 3! So x squared + 5x + 6 = (x + 2)(x + 3). (D) 2x squared + x - 7: with leading coefficient 2 and -7, factoring is complex. Choice C correctly identifies x squared + 5x + 6 as most efficient to factor because it factors easily as (x + 2)(x + 3) = 0, giving x = -2 or x = -3. Choice A would require the quadratic formula since x squared + 2x + 5 doesn't factor over integers (discriminant = 4 - 20 = -16, negative!). Efficiency comes from recognizing factorable patterns: when you see x squared + 5x + 6, immediately think "what multiplies to 6 and adds to 5?" Answer: 2 and 3, so it factors as (x + 2)(x + 3). This takes 5 seconds versus minutes with the quadratic formula!

Question 5

Solve the quadratic equation and express any complex solutions in a±bia \pm bi form:

x2+6x+13=0x^2 + 6x + 13 = 0

  1. x=−3±2ix = -3 \pm 2i (correct answer)
  2. x=3±2ix = 3 \pm 2i
  3. x=−3±4ix = -3 \pm 4i
  4. x=−6±13ix = -6 \pm 13i
Explanation: This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's xx squared = number, inspect or take square roots (x=±numberx = \pm \sqrt{\text{number}}), (2) if it's (expression) squared = number, take square roots, (3) if it factors easily (like xx squared + 5x + 6), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}—it always works!, (5) if asked to derive the formula, complete the square on general form. Matching method to form saves time and reduces errors! For xx squared + 6x + 13 = 0: discriminant 36−52=−1636 - 52 = -16 (negative, so complex), quadratic formula gives x=−6±−162=−6±4i2=−3±2ix = \frac{-6 \pm \sqrt{-16}}{2} = \frac{-6 \pm 4i}{2} = -3 \pm 2i. Choice A correctly applies the formula and simplifies to the a plus or minus bi form. Choice B flips the sign of the real part—remember, it's -b over 2a, so negative for positive b! Complex solution recognition: check discriminant first; negative means i appears. You're mastering complex numbers—keep going!

Question 6

Solve the quadratic equation by factoring: x2−9x+20=0x^2 - 9x + 20 = 0.​

  1. x=−5x=-5 or x=−4x=-4
  2. x=4x=4 only
  3. x=5x=5 or x=4x=4 (correct answer)
  4. x=9x=9 or x=20x=20
Explanation: This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's x squared = number, inspect or take square roots (x = plus or minus square root of number), (2) if it's (expression) squared = number, take square roots, (3) if it factors easily (like x squared + 5x + 6), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula x = (-b plus or minus square root of (b squared - 4ac)) divided by (2a)—it always works!, (5) if asked to derive the formula, complete the square on general form. Matching method to form saves time and reduces errors! For solving x squared - 9x + 20 = 0 by factoring: look for two numbers that multiply to 20 and add to -9, which are -4 and -5, so (x - 4)(x - 5) = 0, then x = 4 or x = 5. Factoring works here because the quadratic factors nicely over integers. Choice B correctly identifies the solutions as x=5 or x=4 by properly factoring and applying the zero product property. Choice C is tempting but fails because it uses positive factors instead of negative, leading to negative solutions which don't satisfy the equation—always check by plugging back in! To master factoring, practice finding factor pairs: for ac=20, pairs are 1 and 20, 2 and 10, 4 and 5—then pick the pair that adds to b with correct signs. Quick method decision: if the constant and coefficient suggest easy factors (like here, 20 and -9), factor first; otherwise, go to quadratic formula for reliability.

Question 7

Solve x2−7x+12=0x^2 - 7x + 12 = 0 by any method. Which set of solutions is correct?

  1. x=3,4x = 3, 4 (correct answer)
  2. x=−3,−4x = -3, -4
  3. x=1,12x = 1, 12
  4. x=7,12x = 7, 12
Explanation: This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's x squared = number, inspect or take square roots (x = plus or minus square root of number), (2) if it's (expression) squared = number, take square roots, (3) if it factors easily (like x squared + 5x + 6), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula x = (-b plus or minus square root of (b squared - 4ac)) divided by (2a)—it always works!, (5) if asked to derive the formula, complete the square on general form. For solving x squared - 7x + 12 = 0, let's try factoring: find two numbers that multiply to 12 and add to -7. Since both must be negative (positive product, negative sum), try: -1 and -12 (add to -13, no), -2 and -6 (add to -8, no), -3 and -4 (add to -7, yes!). So x squared - 7x + 12 = (x - 3)(x - 4) = 0, giving x = 3 or x = 4. Choice A correctly identifies x = 3, 4 as the solutions by factoring (x - 3)(x - 4) = 0. Choice B incorrectly has negative solutions x = -3, -4, which would come from factoring x squared + 7x + 12 = (x + 3)(x + 4) = 0, not our equation. Quick verification: does (x - 3)(x - 4) = x squared - 7x + 12? Expand: x squared - 4x - 3x + 12 = x squared - 7x + 12. Yes! The factoring is correct, giving solutions x = 3 and x = 4. When the constant term is positive and the middle term is negative, both factors have the form (x - positive number).

Question 8

Solve and express the solutions in a±bia\pm bi form: x2+4x+8=0x^2+4x+8=0.

  1. x=−2±2ix=-2\pm 2i (correct answer)
  2. x=2±2ix=2\pm 2i
  3. x=−4±2ix=-4\pm 2i
  4. x=−2±2ix=-2\pm \sqrt{2}i
Explanation: This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's x squared = number, inspect or take square roots (x = plus or minus square root of number), (2) if it's (expression) squared = number, take square roots, (3) if it factors easily (like x squared + 5x + 6), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula x = (-b plus or minus square root of (b squared - 4ac)) divided by (2a)—it always works!, (5) if asked to derive the formula, complete the square on general form. Matching method to form saves time and reduces errors! For solving x² + 4x + 8 = 0: identify a = 1, b = 4, c = 8. Calculate discriminant: b² - 4ac = 16 - 32 = -16 (negative, so complex solutions!). Apply formula: x = (-4 ± √(-16))/2 = (-4 ± 4i)/2 = -2 ± 2i. The negative discriminant signals that i will appear in the answer. Choice A correctly solves to get x = -2 ± 2i by properly applying the quadratic formula and recognizing that √(-16) = 4i. Choice B incorrectly has +2 as the real part instead of -2, forgetting that -b = -4, which when divided by 2a = 2 gives -2, not +2! From formula x = (-b ± √(b² - 4ac))/(2a), separate the real part -b/(2a) from the imaginary part ±√|b² - 4ac|/(2a)·i. Complex solution recognition: For x² + 4x + 8 = 0: discriminant = 16 - 32 = -16 (negative!), so complex solutions: x = (-4 ± 4i)/2 = -2 ± 2i. Quick check: the real part comes from -b/(2a) = -4/2 = -2, and the imaginary part from ±√16/2 = ±2. This systematic approach prevents sign errors!

Question 9

Solve the equation by any method. If the solutions are complex, write them in a±bia\pm bi form:

3x2+6x+5=03x^2+6x+5=0

  1. x=−1±66ix=-1\pm\dfrac{\sqrt{6}}{6}i
  2. x=1±63ix=1\pm\dfrac{\sqrt{6}}{3}i
  3. x=−1±63x=-1\pm\dfrac{\sqrt{6}}{3}
  4. x=−1±63ix=-1\pm\dfrac{\sqrt{6}}{3}i (correct answer)
Explanation: This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's x2=numberx^2 = \text{number}, inspect or take square roots (x=±numberx = \pm \sqrt{\text{number}}), (2) if it's (expression) squared = number, take square roots, (3) if it factors easily (like x2+5x+6x^2 + 5x + 6), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}—it always works!, (5) if asked to derive the formula, complete the square on general form. Matching method to form saves time and reduces errors! For 3x2+6x+5=03x^2 + 6x + 5 = 0: discriminant 36−60=−2436 - 60 = -24 (negative, complex), formula x=−6±−246=−6±26 i6=−1±63 ix = \frac{-6 \pm \sqrt{-24}}{6} = \frac{-6 \pm 2 \sqrt{6} \, i}{6} = -1 \pm \frac{\sqrt{6}}{3} \, i. Choice B correctly computes and simplifies the complex form. Choice A omits i—check discriminant sign! Recognition: negative means a±bia \pm bi. Impressive handling of complexes— you're advancing!

Question 10

Solve by taking square roots: (x−3)2=5(x-3)^2 = 5.

  1. x=3±5x=3\pm\sqrt{5} (correct answer)
  2. x=−3±5x=-3\pm\sqrt{5}
  3. x=3±5x=3\pm 5
  4. x=5−3x=\sqrt{5}-3 only
Explanation: This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's x squared = number, inspect or take square roots (x = plus or minus square root of number), (2) if it's (expression) squared = number, take square roots, (3) if it factors easily (like x squared + 5x + 6), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula x = (-b plus or minus square root of (b squared - 4ac)) divided by (2a)—it always works!, (5) if asked to derive the formula, complete the square on general form. Matching method to form saves time and reduces errors! For (x-3) squared = 5, take square roots on both sides: x-3 = ± sqrt(5), so x = 3 ± sqrt(5)—remember the ± for both solutions. This method is perfect here since it's already in squared form. Choice A correctly applies the square root method, including both positive and negative roots. Choice D fails by omitting the positive root, giving only one solution—quadratics usually have two solutions, so always include ±! Remember, when taking square roots, if the right side is positive, you get two real solutions; if zero, one; if negative, complex. Practice by rewriting equations into squared form when possible for efficiency.

Question 11

Use the quadratic formula to solve:

2x2−3x−5=02x^2-3x-5=0

  1. x=3±494x=\dfrac{3\pm\sqrt{49}}{4} (correct answer)
  2. x=−3±494x=\dfrac{-3\pm\sqrt{49}}{4}
  3. x=3±194x=\dfrac{3\pm\sqrt{19}}{4}
  4. x=3±492x=\dfrac{3\pm\sqrt{49}}{2}
Explanation: This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's x2=numberx^2 = \text{number}, inspect or take square roots (x=±numberx = \pm \sqrt{\text{number}}), (2) if it's (expression)2=number(\text{expression})^2 = \text{number}, take square roots, (3) if it factors easily (like x2+5x+6x^2 + 5x + 6), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}—it always works!, (5) if asked to derive the formula, complete the square on general form. Matching method to form saves time and reduces errors! For 2x2−3x−5=02x^2 - 3x - 5 = 0 using quadratic formula: a=2a=2, b=−3b=-3, c=−5c=-5, discriminant 9+40=499 + 40=49, x=3±74x=\frac{3 \pm 7}{4} (since −b=3-b=3). Choice A correctly plugs in with −b-b positive and denominator 4. Choice B uses +b+b instead of −b-b—double-check the formula sign! Transferable strategy: write aa, bb, cc clearly before plugging in. Great precision—you've got this!

Question 12

Solve by completing the square:

x2+8x+1=0x^2+8x+1=0

  1. x=−4±15x=-4\pm\sqrt{15} (correct answer)
  2. x=4±15x=4\pm\sqrt{15}
  3. x=−8±63x=-8\pm\sqrt{63}
  4. x=−4±17x=-4\pm\sqrt{17}
Explanation: This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's x squared = number, inspect or take square roots (x = plus or minus square root of number), (2) if it's (expression) squared = number, take square roots, (3) if it factors easily (like x squared + 5x + 6), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula x = (-b plus or minus square root of (b squared - 4ac)) divided by (2a)—it always works!, (5) if asked to derive the formula, complete the square on general form. Matching method to form saves time and reduces errors! For x squared + 8x + 1 = 0 by completing square: move 1, add (4)^2=16 to both, (x+4) squared = 15, x = -4 plus or minus sqrt(15). Choice A correctly shifts and takes square root with the right adjustment. Choice B flips the sign—remember, for +bx, you add (b/2)^2 and subtract from x! Strategy: halve b, square it, add to both sides. Super effort—completing the square is versatile!

Question 13

Solve by completing the square: x2+6x+2=0x^2 + 6x + 2 = 0.​

  1. x=3±7x=3\pm\sqrt{7}
  2. x=−3±11x=-3\pm\sqrt{11}
  3. x=−6±2x=-6\pm\sqrt{2}
  4. x=−3±7x=-3\pm\sqrt{7} (correct answer)
Explanation: This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's x squared = number, inspect or take square roots (x = plus or minus square root of number), (2) if it's (expression) squared = number, take square roots, (3) if it factors easily (like x squared + 5x + 6), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula x = (-b plus or minus square root of (b squared - 4ac)) divided by (2a)—it always works!, (5) if asked to derive the formula, complete the square on general form. Matching method to form saves time and reduces errors! For x squared + 6x + 2 = 0, move 2: x squared + 6x = -2; half of 6 is 3, square to 9; add 9: (x+3) squared = 7; x+3 = ± sqrt(7); x = -3 ± sqrt(7). This method avoids fractions initially. Choice A correctly completes the square and isolates x. Choice B reverses the sign on the real part—remember it's - (half of b) for the shift! Completing the square strategy: always works, especially when a=1 and b even. Use it to verify formula results too.

Question 14

Which quadratic equation is most efficiently solved by factoring?

  1. x2+5x+1=0x^2+5x+1=0
  2. x2+3x−1=0x^2+3x-1=0
  3. x2+5x+6=0x^2+5x+6=0 (correct answer)
  4. 3x2−5x+1=03x^2-5x+1=0
Explanation: This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's x squared = number, inspect or take square roots (x = plus or minus square root of number), (2) if it's (expression) squared = number, take square roots, (3) if it factors easily (like x squared + 5x + 6), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula x = (-b plus or minus square root of (b squared - 4ac)) divided by (2a)—it always works!, (5) if asked to derive the formula, complete the square on general form. Matching method to form saves time and reduces errors! To determine which factors easily, check if you can find two numbers that multiply to c and add to b in under 5 seconds. For x² + 5x + 6: need numbers that multiply to 6 and add to 5—that's 2 and 3! So x² + 5x + 6 = (x + 2)(x + 3). For x² + 5x + 1: no integer pairs multiply to 1 and add to 5. Choice C (x² + 5x + 6) is most efficiently solved by factoring because 2 and 3 multiply to 6 and add to 5, giving (x + 2)(x + 3) = 0 instantly. Choice A (x² + 5x + 1) doesn't factor over integers—you'd need the quadratic formula, making it less efficient for factoring! Quick method decision tree: DOES it factor in 5 seconds mentally? For x² + 5x + 6, we need 2 and 3 (multiply to 6, add to 5)—factors instantly as (x + 2)(x + 3)! For x² + 5x + 1, no integer factors work—use quadratic formula instead. Efficiency comes from matching method to form!

Question 15

Solve by completing the square: x2+6x+1=0.x^2+6x+1=0.

  1. x=−3±8x=-3\pm\sqrt{8}
  2. x=3±8x=3\pm\sqrt{8}
  3. x=−3±22x=-3\pm 2\sqrt{2} (correct answer)
  4. x=−6±35x=-6\pm\sqrt{35}
Explanation: This question targets completing the square, a method that rewrites quadratics as (x+h)2=k(x + h)^2 = k, then takes square roots—useful for deriving the formula or when factoring fails. For x2+6x+1=0x^2 + 6x + 1 = 0, move 1: x2+6x=−1x^2 + 6x = -1, add (62)2=9(\frac{6}{2})^2 = 9: (x+3)2=8(x+3)^2 = 8, x+3=±8=±22x+3 = \pm \sqrt{8} = \pm 2\sqrt{2}, x=−3±22x = -3 \pm 2\sqrt{2}—perfect! This method shines for monic quadratics without even coefficients. Choice C correctly simplifies 8\sqrt{8} to 222\sqrt{2}. Choice A forgets to simplify the radical—always reduce for clean answers. Practice by halving b and squaring to complete efficiently. You've got the hang of it—keep building!

Question 16

Solve the equation and write the solutions in a±bia\pm bi form: x2+4x+8=0x^2 + 4x + 8 = 0.​

  1. x=−2±2ix=-2\pm 2i (correct answer)
  2. x=−2±4ix=-2\pm 4i
  3. x=−4±2ix=-4\pm 2i
  4. x=2±2ix=2\pm 2i
Explanation: This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's x squared = number, inspect or take square roots (x = plus or minus square root of number), (2) if it's (expression) squared = number, take square roots, (3) if it factors easily (like x squared + 5x + 6), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula x = (-b plus or minus square root of (b squared - 4ac)) divided by (2a)—it always works!, (5) if asked to derive the formula, complete the square on general form. Matching method to form saves time and reduces errors! For x squared + 4x + 8 = 0, discriminant = 16 - 32 = -16 (negative), so complex: using quadratic formula, x = [-4 ± sqrt(-16)] / 2 = [-4 ± 4i] / 2 = -2 ± 2i. Completing the square also works: x squared + 4x = -8, add 4: (x+2) squared = -4, x+2 = ± 2i, x = -2 ± 2i. Choice A correctly expresses the complex solutions in a ± bi form after proper calculation. Choice C is tempting but doubles the imaginary part incorrectly—remember to divide the entire numerator by 2a. Complex solution tip: factor out the square root of -1 as i, and simplify fractions. Always check discriminant first to predict solution type.

Question 17

Solve using the quadratic formula: 2x2+3x−7=02x^2 + 3x - 7 = 0.

  1. x=−3±654x = \dfrac{-3 \pm \sqrt{65}}{4} (correct answer)
  2. x=3±654x = \dfrac{3 \pm \sqrt{65}}{4}
  3. x=−3±734x = \dfrac{-3 \pm \sqrt{73}}{4}
  4. x=−3±652x = \dfrac{-3 \pm \sqrt{65}}{2}
Explanation: This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's x squared = number, inspect or take square roots (x = plus or minus square root of number), (2) if it's (expression) squared = number, take square roots, (3) if it factors easily (like x squared + 5x + 6), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula x = (-b plus or minus square root of (b squared - 4ac)) divided by (2a)—it always works!, (5) if asked to derive the formula, complete the square on general form. For solving 2x squared + 3x - 7 = 0 by quadratic formula: identify a = 2, b = 3, c = -7. Calculate discriminant: b squared - 4ac = 9 - 4(2)(-7) = 9 + 56 = 65 (positive, so two real solutions). Apply formula: x = (-3 plus or minus square root of 65) divided by (2 times 2) = (-3 plus or minus square root of 65) divided by 4. Choice A correctly applies the quadratic formula with x = (-3 plus or minus square root of 65) divided by 4. Choice D incorrectly has denominator 2 instead of 2a = 4—remember the denominator is 2a, not just 2! When a = 2, the denominator becomes 2(2) = 4. This is a common error: forgetting to multiply by the coefficient 'a' in the denominator. Always use 2a in the denominator, where a is the coefficient of x squared. For 2x squared + 3x - 7 = 0, we have a = 2, so denominator = 2(2) = 4, giving x = (-3 plus or minus square root of 65) divided by 4.

Question 18

Predict the solution type using the discriminant, then solve in a±bia\pm bi form: x2−6x+10=0x^2 - 6x + 10 = 0.

  1. x=3±ix=3\pm i (correct answer)
  2. x=3±2ix=3\pm 2i
  3. x=−3±ix=-3\pm i
  4. x=3±2x=3\pm\sqrt{2}
Explanation: This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's x squared = number, inspect or take square roots (x = plus or minus square root of number), (2) if it's (expression) squared = number, take square roots, (3) if it factors easily (like x squared + 5x + 6), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula x = (-b plus or minus square root of (b squared - 4ac)) divided by (2a)—it always works!, (5) if asked to derive the formula, complete the square on general form. Matching method to form saves time and reduces errors! For x squared - 6x + 10 = 0, discriminant=36-40=-4 (negative, complex solutions); x=[6 ± sqrt(-4)]/2 = [6 ± 2i]/2 = 3 ± i. Predicting with discriminant guides the method. Choice A correctly predicts complex type and solves accurately. Choice D assumes real solutions, ignoring negative discriminant—always calculate it first! Discriminant strategy: positive=two real, zero=one real, negative=two complex. This quick check informs your approach.

Question 19

Solve by taking square roots: (x−3)2=5(x-3)^2 = 5.​

  1. x=3±5x=3\pm\sqrt{5} (correct answer)
  2. x=−3±5x=-3\pm\sqrt{5}
  3. x=3±5x=3\pm 5
  4. x=5−3x=\sqrt{5}-3 only
Explanation: This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's x squared = number, inspect or take square roots (x = plus or minus square root of number), (2) if it's (expression) squared = number, take square roots, (3) if it factors easily (like x squared + 5x + 6), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula x = (-b plus or minus square root of (b squared - 4ac)) divided by (2a)—it always works!, (5) if asked to derive the formula, complete the square on general form. Matching method to form saves time and reduces errors! For (x-3) squared = 5, take square roots on both sides: x-3 = ± sqrt(5), so x = 3 ± sqrt(5)—remember the ± for both solutions. This method is perfect here since it's already in squared form. Choice A correctly applies the square root method, including both positive and negative roots. Choice D fails by omitting the positive root, giving only one solution—quadratics usually have two solutions, so always include ±! Remember, when taking square roots, if the right side is positive, you get two real solutions; if zero, one; if negative, complex. Practice by rewriting equations into squared form when possible for efficiency.

Question 20

Solve using the quadratic formula: 2x2−3x−7=02x^2-3x-7=0.

  1. x=3±654x=\dfrac{3\pm\sqrt{65}}{4} (correct answer)
  2. x=−3±654x=\dfrac{-3\pm\sqrt{65}}{4}
  3. x=3±174x=\dfrac{3\pm\sqrt{17}}{4}
  4. x=3±652x=\dfrac{3\pm\sqrt{65}}{2}
Explanation: This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's x squared = number, inspect or take square roots (x = plus or minus square root of number), (2) if it's (expression) squared = number, take square roots, (3) if it factors easily (like x squared + 5x + 6), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula x = (-b plus or minus square root of (b squared - 4ac)) divided by (2a)—it always works!, (5) if asked to derive the formula, complete the square on general form. Matching method to form saves time and reduces errors! For solving 2x² - 3x - 7 = 0 by quadratic formula: identify a = 2, b = -3, c = -7. Calculate discriminant: b² - 4ac = (-3)² - 4(2)(-7) = 9 + 56 = 65 (positive, so two real solutions!). Apply formula: x = (3 ± √65)/(2·2) = (3 ± √65)/4. Note that -b = -(-3) = 3, which is positive! Choice A correctly applies the quadratic formula with proper sign handling: -b = -(-3) = +3, giving x = (3 ± √65)/4. Choice B incorrectly uses -3 instead of 3 in the numerator, forgetting that when b = -3, then -b = +3—a common sign error in the quadratic formula! Quick method decision tree: For 2x² - 3x - 7 = 0, factoring would be messy (need to factor 2·(-7) = -14), so jump straight to the quadratic formula. With a = 2, b = -3, c = -7, we get x = (3 ± √65)/4. The formula always works and gives exact answers!