Praxis Math Quiz: Manipulate Algebraic Expressions
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Manipulate Algebraic ExpressionsQuestion 1 of 20

A large rectangular field has a length of 5x+45x+4 and a width of 2x−12x-1. A smaller rectangular patch within the field has dimensions x+2x+2 and x−1x-1. Which expression represents the area of the field that is outside the smaller patch?

9x2+x−69x^2 + x - 6
9x2+2x−29x^2 + 2x - 2
11x2+4x−211x^2 + 4x - 2
9x2+2x−69x^2 + 2x - 6
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Praxis Math Quiz: Manipulate Algebraic Expressions

Practice Manipulate Algebraic Expressions in Praxis Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

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This quiz focuses on Manipulate Algebraic Expressions, giving you a quick way to practice the rules, question types, and explanations that matter most for Praxis Math.

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Question 1

A large rectangular field has a length of 5x+45x+4 and a width of 2x−12x-1. A smaller rectangular patch within the field has dimensions x+2x+2 and x−1x-1. Which expression represents the area of the field that is outside the smaller patch?

  1. 9x2+x−69x^2 + x - 6
  2. 9x2+2x−29x^2 + 2x - 2 (correct answer)
  3. 11x2+4x−211x^2 + 4x - 2
  4. 9x2+2x−69x^2 + 2x - 6
Explanation: First, find the area of the large field: Alarge=(5x+4)(2x−1)=10x2−5x+8x−4=10x2+3x−4A_{large} = (5x+4)(2x-1) = 10x^2 - 5x + 8x - 4 = 10x^2 + 3x - 4. Next, find the area of the smaller patch: Asmall=(x+2)(x−1)=x2−x+2x−2=x2+x−2A_{small} = (x+2)(x-1) = x^2 - x + 2x - 2 = x^2 + x - 2. To find the remaining area, subtract the small area from the large area: (10x2+3x−4)−(x2+x−2)(10x^2 + 3x - 4) - (x^2 + x - 2). Distribute the negative sign: 10x2+3x−4−x2−x+210x^2 + 3x - 4 - x^2 - x + 2. Combine like terms: (10x2−x2)+(3x−x)+(−4+2)(10x^2-x^2) + (3x-x) + (-4+2), which simplifies to 9x2+2x−29x^2 + 2x - 2.

Question 2

Which of the following is equivalent to 2x−1+3x+1\frac{2}{x-1} + \frac{3}{x+1}?

  1. 5x−1x2−1\frac{5x-1}{x^2-1} (correct answer)
  2. 52x\frac{5}{2x}
  3. 5x+1x2−1\frac{5x+1}{x^2-1}
  4. 5x2−1\frac{5}{x^2-1}
Explanation: To add these fractions, find a common denominator, which is (x−1)(x+1)(x-1)(x+1) or x2−1x^2-1. Rewrite each fraction with the common denominator: 2(x+1)(x−1)(x+1)+3(x−1)(x−1)(x+1)\frac{2(x+1)}{(x-1)(x+1)} + \frac{3(x-1)}{(x-1)(x+1)}. Now, add the numerators: 2(x+1)+3(x−1)x2−1\frac{2(x+1) + 3(x-1)}{x^2-1}. Distribute in the numerator: 2x+2+3x−3x2−1\frac{2x+2+3x-3}{x^2-1}. Combine like terms in the numerator: 5x−1x2−1\frac{5x-1}{x^2-1}.

Question 3

Which expression is equivalent to ab−5a+2b−10ab - 5a + 2b - 10?

  1. (a+2)(b−5)(a+2)(b-5) (correct answer)
  2. (a−2)(b+5)(a-2)(b+5)
  3. (a+5)(b−2)(a+5)(b-2)
  4. (a−5)(b−2)(a-5)(b-2)
Explanation: This expression can be factored by grouping. Group the first two terms and the last two terms: (ab−5a)+(2b−10)(ab - 5a) + (2b - 10). Factor out the greatest common factor from each group: a(b−5)+2(b−5)a(b - 5) + 2(b - 5). Now, factor out the common binomial factor (b−5)(b - 5): (a+2)(b−5)(a + 2)(b - 5).

Question 4

If a−b=5a - b = 5 and a+b=11a + b = 11, what is the value of the expression a2−b2a^2 - b^2?

  1. 1616
  2. 9696
  3. 66
  4. 5555 (correct answer)
Explanation: This problem can be solved without finding the individual values of aa and bb. The expression a2−b2a^2 - b^2 is a difference of squares, which factors into (a−b)(a+b)(a - b)(a + b). We are given the values of these two factors: a−b=5a - b = 5 and a+b=11a + b = 11. Therefore, a2−b2=(5)(11)=55a^2 - b^2 = (5)(11) = 55.

Question 5

The area of a square is given by the expression 9x2+30x+259x^2 + 30x + 25. Which expression represents the length of one side of the square?

  1. 3x−53x - 5
  2. 9x+59x + 5
  3. 9x+259x + 25
  4. 3x+53x + 5 (correct answer)
Explanation: The area of a square is the side length squared (A=s2A = s^2). To find the side length, we need to find the square root of the area expression, which means we need to factor the perfect square trinomial 9x2+30x+259x^2 + 30x + 25. A perfect square trinomial has the form a2+2ab+b2=(a+b)2a^2 + 2ab + b^2 = (a+b)^2. Here, a2=9x2a^2 = 9x^2 so a=3xa=3x, and b2=25b^2 = 25 so b=5b=5. Check the middle term: 2ab=2(3x)(5)=30x2ab = 2(3x)(5) = 30x. This matches. Thus, the factored form is (3x+5)2(3x+5)^2. The length of one side is 3x+53x+5.

Question 6

Which expression represents the product of (x−y+2)(x - y + 2) and (x−y−2)(x - y - 2)?

  1. x2−y2−4x^2 - y^2 - 4
  2. x2−2xy+y2−4x^2 - 2xy + y^2 - 4 (correct answer)
  3. x2+2xy+y2−4x^2 + 2xy + y^2 - 4
  4. x2−y2+4x^2 - y^2 + 4
Explanation: This problem can be simplified by recognizing a difference of squares pattern. Let A=(x−y)A = (x-y). The expression becomes (A+2)(A−2)(A+2)(A-2), which simplifies to A2−22=A2−4A^2 - 2^2 = A^2 - 4. Now, substitute (x−y)(x-y) back in for AA: (x−y)2−4(x-y)^2 - 4. Expanding (x−y)2(x-y)^2 gives x2−2xy+y2x^2 - 2xy + y^2. So the final expression is x2−2xy+y2−4x^2 - 2xy + y^2 - 4.

Question 7

Simplify the expression xy−1x2y2−1\frac{\frac{x}{y} - 1}{\frac{x^2}{y^2} - 1}.

  1. yx+y\frac{y}{x+y} (correct answer)
  2. x−yx+y\frac{x-y}{x+y}
  3. yx−y\frac{y}{x-y}
  4. xy+1\frac{x}{y+1}
Explanation: To simplify this complex fraction, first rewrite the numerator and denominator with common denominators. Numerator: xy−yy=x−yy\frac{x}{y} - \frac{y}{y} = \frac{x-y}{y}. Denominator: x2y2−y2y2=x2−y2y2\frac{x^2}{y^2} - \frac{y^2}{y^2} = \frac{x^2-y^2}{y^2}. The expression is now x−yyx2−y2y2\frac{\frac{x-y}{y}}{\frac{x^2-y^2}{y^2}}. To divide by a fraction, multiply by its reciprocal: x−yy⋅y2x2−y2\frac{x-y}{y} \cdot \frac{y^2}{x^2-y^2}. Factor the difference of squares: x−yy⋅y2(x−y)(x+y)\frac{x-y}{y} \cdot \frac{y^2}{(x-y)(x+y)}. Cancel the common factor (x−y)(x-y) and one yy. The result is yx+y\frac{y}{x+y}.

Question 8

The sum of the squares of two consecutive odd integers is represented by the expression (2n+1)2+(2n+3)2(2n+1)^2 + (2n+3)^2 for some integer nn. Which of the following is a simplified form of this expression?

  1. 8n2+16n+108n^2 + 16n + 10 (correct answer)
  2. 8n2+108n^2 + 10
  3. 4n2+8n+104n^2 + 8n + 10
  4. 8n2+8n+108n^2 + 8n + 10
Explanation: First, expand each squared binomial. (2n+1)2=(2n+1)(2n+1)=4n2+2n+2n+1=4n2+4n+1(2n+1)^2 = (2n+1)(2n+1) = 4n^2 + 2n + 2n + 1 = 4n^2 + 4n + 1. (2n+3)2=(2n+3)(2n+3)=4n2+6n+6n+9=4n2+12n+9(2n+3)^2 = (2n+3)(2n+3) = 4n^2 + 6n + 6n + 9 = 4n^2 + 12n + 9. Now, add the two expanded expressions: (4n2+4n+1)+(4n2+12n+9)(4n^2 + 4n + 1) + (4n^2 + 12n + 9). Combine like terms: (4n2+4n2)+(4n+12n)+(1+9)(4n^2+4n^2) + (4n+12n) + (1+9), which simplifies to 8n2+16n+108n^2 + 16n + 10.

Question 9

If x2+kx−18x^2 + kx - 18 can be factored into (x−3)(x+6)(x-3)(x+6), what is the value of kk?

  1. −3-3
  2. 33 (correct answer)
  3. −9-9
  4. 99
Explanation: To find the value of kk, we need to expand the factored form (x−3)(x+6)(x-3)(x+6) and compare it to the original trinomial x2+kx−18x^2 + kx - 18. Using the FOIL method on (x−3)(x+6)(x-3)(x+6): First terms x⋅x=x2x \cdot x = x^2, Outer terms x⋅6=6xx \cdot 6 = 6x, Inner terms −3⋅x=−3x-3 \cdot x = -3x, and Last terms −3⋅6=−18-3 \cdot 6 = -18. Combining these gives x2+6x−3x−18x^2 + 6x - 3x - 18, which simplifies to x2+3x−18x^2 + 3x - 18. Comparing this to x2+kx−18x^2 + kx - 18, we can see that kk must be 3.

Question 10

When 5a3−45ab25a^3 - 45ab^2 is factored completely, which of the following is one of its factors?

  1. (a−9b)(a - 9b)
  2. (a−3b)(a - 3b) (correct answer)
  3. 55
  4. (a2−9b2)(a^2 - 9b^2)
Explanation: First, identify the greatest common factor (GCF) of the terms. Both terms are divisible by 5a5a. Factoring out 5a5a gives 5a(a2−9b2)5a(a^2 - 9b^2). The term in the parentheses, a2−9b2a^2 - 9b^2, is a difference of squares, a2−(3b)2a^2 - (3b)^2, which factors into (a−3b)(a+3b)(a - 3b)(a + 3b). Therefore, the completely factored expression is 5a(a−3b)(a+3b)5a(a - 3b)(a + 3b). One of the factors is (a−3b)(a - 3b).

Question 11

Which of the following expressions is equivalent to 18x5−12x4+6x26x2\frac{18x^5 - 12x^4 + 6x^2}{6x^2} for x≠0x \neq 0?

  1. 3x3−2x23x^3 - 2x^2
  2. 12x3−6x2+112x^3 - 6x^2 + 1
  3. 3x3−2x2+13x^3 - 2x^2 + 1 (correct answer)
  4. 3x3−2x2+x3x^3 - 2x^2 + x
Explanation: To simplify, divide each term in the numerator by the monomial in the denominator. 18x56x2−12x46x2+6x26x2\frac{18x^5}{6x^2} - \frac{12x^4}{6x^2} + \frac{6x^2}{6x^2}. For the first term, 18/6=318/6 = 3 and x5/x2=x5−2=x3x^5/x^2 = x^{5-2} = x^3. For the second term, 12/6=212/6 = 2 and x4/x2=x4−2=x2x^4/x^2 = x^{4-2} = x^2. For the third term, 6x2/6x2=16x^2/6x^2 = 1. Combining these results gives 3x3−2x2+13x^3 - 2x^2 + 1.

Question 12

The expression (3x4y−2)3(3x^4y^{-2})^3 is equivalent to which of the following?

  1. 9x12y6\frac{9x^{12}}{y^6}
  2. 9x7y9x^7y
  3. 27x12y6\frac{27x^{12}}{y^6} (correct answer)
  4. 27x7y−1\frac{27x^7}{y^{-1}}
Explanation: To simplify the expression, apply the exponent of 3 to each factor inside the parentheses. This gives 33⋅(x4)3⋅(y−2)33^3 \cdot (x^4)^3 \cdot (y^{-2})^3. Calculate each part: 33=273^3 = 27, (x4)3=x4⋅3=x12(x^4)^3 = x^{4 \cdot 3} = x^{12}, and (y−2)3=y−2⋅3=y−6(y^{-2})^3 = y^{-2 \cdot 3} = y^{-6}. The expression is 27x12y−627x^{12}y^{-6}. A negative exponent indicates the reciprocal, so y−6=1y6y^{-6} = \frac{1}{y^6}. The final simplified expression is 27x12y6\frac{27x^{12}}{y^6}.

Question 13

The volume of a rectangular prism is given by the expression x3+7x2+10xx^3 + 7x^2 + 10x. If the height of the prism is xx, which of the following could represent the length and width of the prism's base?

  1. xx and x+7x+7
  2. x+2x+2 and x+5x+5 (correct answer)
  3. x+7x+7 and x+10x+10
  4. x−2x-2 and x−5x-5
Explanation: The volume of a prism is V=length×width×heightV = \text{length} \times \text{width} \times \text{height}. We are given V=x3+7x2+10xV = x^3 + 7x^2 + 10x and height h=xh=x. The area of the base is V/hV/h. First, factor out the GCF from the volume expression, which is xx. This gives V=x(x2+7x+10)V = x(x^2 + 7x + 10). Since the height is xx, the area of the base must be x2+7x+10x^2 + 7x + 10. To find the length and width, we factor this trinomial. We need two numbers that multiply to 10 and add to 7. These numbers are 2 and 5. So, the base area factors to (x+2)(x+5)(x+2)(x+5). These are the expressions for the length and width.

Question 14

Which of the following expressions is equivalent to (64a6)1/2−(8a9)1/3(64a^6)^{1/2} - (8a^9)^{1/3} for a≥0a \ge 0?

  1. 6a36a^3 (correct answer)
  2. 30a330a^3
  3. 6a6a
  4. 8a3−2a68a^3 - 2a^6
Explanation: Simplify each term separately. For the first term: (64a6)1/2=64⋅a6=8⋅a3=8a3(64a^6)^{1/2} = \sqrt{64} \cdot \sqrt{a^6} = 8 \cdot a^3 = 8a^3. For the second term: (8a9)1/3=83⋅a93=2⋅a3=2a3(8a^9)^{1/3} = \sqrt[3]{8} \cdot \sqrt[3]{a^9} = 2 \cdot a^3 = 2a^3. The expression becomes 8a3−2a38a^3 - 2a^3. Combine like terms: 8a3−2a3=6a38a^3 - 2a^3 = 6a^3.

Question 15

Which of the following is equivalent to 3x+9x2÷x2−9x\frac{3x+9}{x^2} \div \frac{x^2-9}{x} for all valid values of xx?

  1. 3x(x+3)\frac{3}{x(x+3)}
  2. 3(x+3)2x3\frac{3(x+3)^2}{x^3}
  3. 3x(x−3)\frac{3}{x(x-3)} (correct answer)
  4. 3(x−3)x\frac{3(x-3)}{x}
Explanation: To divide by a fraction, we multiply by its reciprocal. The expression becomes 3x+9x2⋅xx2−9\frac{3x+9}{x^2} \cdot \frac{x}{x^2-9}. Next, factor each polynomial. 3x+9=3(x+3)3x+9 = 3(x+3) and x2−9=(x−3)(x+3)x^2-9 = (x-3)(x+3). Substituting these back into the expression gives 3(x+3)x2⋅x(x−3)(x+3)\frac{3(x+3)}{x^2} \cdot \frac{x}{(x-3)(x+3)}. Now, cancel common factors. The (x+3)(x+3) in the numerator cancels with the (x+3)(x+3) in the denominator. One xx in the numerator cancels with one xx in the denominator's x2x^2. This leaves 3x(x−3)\frac{3}{x(x-3)}.

Question 16

The expression x4−81x^4 - 81 is equivalent to which of the following?

  1. (x2−9)(x2−9)(x^2 - 9)(x^2 - 9)
  2. (x−3)4(x - 3)^4
  3. (x2+9)(x−3)(x+3)(x^2 + 9)(x - 3)(x + 3) (correct answer)
  4. (x−3)(x+3)(x−3)(x+3)(x - 3)(x + 3)(x - 3)(x + 3)
Explanation: The expression x4−81x^4 - 81 is a difference of squares, where the terms are (x2)2(x^2)^2 and 929^2. It factors into (x2−9)(x2+9)(x^2 - 9)(x^2 + 9). The term x2+9x^2 + 9 is a sum of squares and cannot be factored further using real numbers. The term x2−9x^2 - 9 is another difference of squares, which factors into (x−3)(x+3)(x - 3)(x + 3). Therefore, the completely factored expression is (x−3)(x+3)(x2+9)(x - 3)(x + 3)(x^2 + 9).

Question 17

The expression (x+y)−2(x+y)^{-2} is equivalent to which of the following?

  1. x−2+y−2x^{-2} + y^{-2}
  2. 1(x+y)2\frac{1}{(x+y)^2} (correct answer)
  3. 1x2+1y2\frac{1}{x^2} + \frac{1}{y^2}
  4. −2(x+y)-2(x+y)
Explanation: A negative exponent indicates the reciprocal of the base raised to the positive value of the exponent. In this case, the base is (x+y)(x+y) and the exponent is −2-2. Therefore, (x+y)−2=1(x+y)2(x+y)^{-2} = \frac{1}{(x+y)^2}. It is a common mistake to distribute the exponent to each term inside the parenthesis, but this is incorrect.

Question 18

Simplify the expression 16x2+9x2\sqrt{16x^2} + \sqrt{9x^2} for x≥0x \ge 0.

  1. 25x2\sqrt{25x^2}
  2. 7x7x (correct answer)
  3. 7x27x^2
  4. 5x5x
Explanation: First, simplify each square root term individually. 16x2=16⋅x2=4x\sqrt{16x^2} = \sqrt{16} \cdot \sqrt{x^2} = 4x (since x≥0x \ge 0). Similarly, 9x2=9⋅x2=3x\sqrt{9x^2} = \sqrt{9} \cdot \sqrt{x^2} = 3x. The expression becomes 4x+3x4x + 3x. Since these are like terms, they can be added together to get 7x7x. It is incorrect to add the numbers under the radicals first.

Question 19

The perimeter of a triangle is given by the expression 7x−27x-2. If the lengths of two sides are 2x+52x+5 and 3x−43x-4, what is the length of the third side?

  1. 2x−32x - 3 (correct answer)
  2. 12x−112x - 1
  3. 2x+12x + 1
  4. 5x+15x + 1
Explanation: The perimeter of a triangle is the sum of its three sides. Let the third side be SS. Then P=(2x+5)+(3x−4)+SP = (2x+5) + (3x-4) + S. We are given P=7x−2P = 7x-2. First, find the sum of the two known sides: (2x+5)+(3x−4)=5x+1(2x+5) + (3x-4) = 5x+1. Now, subtract this sum from the total perimeter to find the length of the third side: S=(7x−2)−(5x+1)S = (7x-2) - (5x+1). Distribute the negative: S=7x−2−5x−1S = 7x - 2 - 5x - 1. Combine like terms: S=(7x−5x)+(−2−1)=2x−3S = (7x-5x) + (-2-1) = 2x-3.

Question 20

Which expression is equivalent to (x2+3x−1)−(x2−x+4)+(2x2+5)(x^2+3x-1) - (x^2-x+4) + (2x^2+5)?

  1. 2x2+2x2x^2 + 2x
  2. 4x2+2x+84x^2 + 2x + 8
  3. 2x2+4x2x^2 + 4x (correct answer)
  4. 2x2+4x+82x^2 + 4x + 8
Explanation: To simplify, first remove the parentheses, being careful to distribute the negative sign. The expression becomes x2+3x−1−x2+x−4+2x2+5x^2+3x-1 - x^2+x-4 + 2x^2+5. Now, group and combine like terms. For x2x^2 terms: x2−x2+2x2=2x2x^2 - x^2 + 2x^2 = 2x^2. For xx terms: 3x+x=4x3x + x = 4x. For constant terms: −1−4+5=0-1 - 4 + 5 = 0. The simplified expression is 2x2+4x2x^2 + 4x.