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ACT Math : Squaring / Square Roots / Radicals

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Example Questions

Example Question #1 : How To Find The Square Of Difference

$\small (3a-2b)^{2}$ can be rewritten as:

Possible Answers:

$\small 9a^{2}-12ab+4b^{2}$

$\small 9a^{2}-4b^{2}$

$\small 9a^{2}-6ab+4b^{2}$

$\small 9a^{2}+4b^{2}$

$\small 9a^{2}+12ab+4b^{2}$

Correct answer:

$\small 9a^{2}-12ab+4b^{2}$

Explanation:

Use the formula for solving the square of a difference, $\small (x-y)^{2}=x^{2}-2xy+y^{2}$ . In this case, $\small \small \small (3a-2b)^{2}=9a^{2}-12ab+4b^{2}$

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Example Question #1 : How To Find The Square Of Difference

Expand:

(2x-6)^2

Possible Answers:

4x-12

4x^2-12x+36

4x^2-24x+36

2x^2-12x+24

Correct answer:

4x^2-24x+36

Explanation:

To multiply a difference squared, square the first term and add two times the multiplication of the two terms. Then add the second term squared.

(2x^)^2+2(2x)(-6)+(-6)^2=4x^2-24x+36

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Example Question #1 : How To Find The Square Of Difference

The expression $\frac{x^2-25}{x^2+11x+30}$ is equivalent to:

Possible Answers:

x+6

$\frac{x+5}{x-6}$

$\frac{x+6}{x+5}$

$\frac{x-5}{x+6}$

$\frac{x-6}{x-5}$

Correct answer:

$\frac{x-5}{x+6}$

Explanation:

First, we need to factor the numerator and denominator separately and cancel out similar terms. We will start with the numerator because it can be factored easily as the difference of two squares.

x^2 - 25 = (x + 5)(x - 5)

Now factor the quadratic in the denominator.

x^2 + 11x + 30 = (x + 5)(x + 6)

Substitute these factorizations back into the original expression.

$\frac{(x+5)(x-5)}{(x+5)(x+6)}$

The (x+5) terms cancel out, leaving us with the following answer:

$\frac{x-5}{x+6}$

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Example Question #113 : Exponents

Evaluate the following expression:

$2^{5}+(5*2)^{2}$

Possible Answers:

132

110

164

Correct answer:

132

Explanation:

2 raised to the power of 5 is the same as multiplying 2 by itself 5 times so:

2⁵= 2x2x2x2x2 = 32

Then, 5x2 must first be multiplied before taking the exponent, yielding 10²= 100.

100 + 32 = 132

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Example Question #2 : Square Of Sum

Expand:

(3x+5)^2

Possible Answers:

9x^2+30x+25

9x^2-30x+25

9x^2+25

6x+10

Correct answer:

9x^2+30x+25

Explanation:

To multiply a difference squared, square the first term and add two times the multiplication of the two terms. Then add the second term squared.

(3x)^2+2(3x)(5)+(5)^2=9x^2+30x+25

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Example Question #1 : How To Find The Square Of A Sum

Which of the following is the square of 4x + 7y ?

Possible Answers:

$16x^{2}+ 22xy+ 49y^{2}$

$16x^{2}+ 28xy+ 49y^{2}$

$16x^{2}+ 56xy+ 49y^{2}$

$16x^{2} + 49y^{2}$

$16x^{2}+ 11xy+ 49y^{2}$

Correct answer:

$16x^{2}+ 56xy+ 49y^{2}$

Explanation:

Use the square of a sum pattern, substituting for and for in the pattern:

$(A+ B)^{2}= A^{2}+ 2AB + B^{2}$

$(4x + 7y)^{2}= (4x)^{2}+ 2 (4x) (7y) + (7y)^{2}$

$= 16x^{2}+ 56xy + 49y^{2}$

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Example Question #2 : Squaring / Square Roots / Radicals

Which of the following is the square of $5 \sqrt{x} +3 \sqrt{y}$ ?

You may assume both and are positive.

Possible Answers:

$25x + 9y+ 15\sqrt{xy}$

25x + 9y

$25x + 9y+ 30 \sqrt{xy}$

$5x + 3y+ 30\sqrt{xy}$

$5x + 3y+ 15\sqrt{xy}$

Correct answer:

$25x + 9y+ 30 \sqrt{xy}$

Explanation:

Use the square of a sum pattern, substituting $5 \sqrt{x}$ for and $3 \sqrt{y}$ for in the pattern:

$(A+ B)^{2}= A^{2}+ 2AB + B^{2}$

$(5 \sqrt{x}+3 \sqrt{y})^{2}= (5 \sqrt{x})^{2}+ 2 (5 \sqrt{x})\left (3 \sqrt{y} \right ) + \left (3 \sqrt{y} \right )^{2}$

$(5 \sqrt{x}+3 \sqrt{y})^{2}= 25x+ 30 \sqrt{xy} + 9y$

$25x + 9y+ 30 \sqrt{xy}$

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Example Question #4 : Squaring / Square Roots / Radicals

Which of the following is the square of $x^{2}+ x + 7$ ?

Possible Answers:

$x^{4}+2x^{3}+8x^{2}+8x+49$

$x^{4}+ x^{3}+14x^{2}+7x+49$

$x^{4}+2x^{3}+15x^{2}+14x+49$

$x^{4}+ x^{2}+ 49$

$x^{4}+ x^{3}+14x^{2}+14x+49$

Correct answer:

$x^{4}+2x^{3}+15x^{2}+14x+49$

Explanation:

Multiply vertically as follows:

$\begin{matrix} x^{2}+ \; \; \; x +\; \; \; 7\\\underline{x^{2}+\; \; \; x +\; \; \; 7} \end{matrix}$

$7 x^{2}+7 x +49$

$x^{3}+ x^{2} + 7x$

$\underline{x^{4}+ x^{3}+7 x^{2} \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; }$

$x^{4}+2x^{3}+15x^{2}+14x+49$

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Example Question #6 : How To Find The Square Of A Sum

Which of the following is the square of $y + \sqrt{17}$ ?

Possible Answers:

$y^{2}+ y\sqrt{17}+17$

$y^{2}+ 34y+17$

$y^{2}+ 17y+17$

The correct answer is not given among the other responses.

$y^{2}+ y\sqrt{34}+17$

Correct answer:

The correct answer is not given among the other responses.

Explanation:

Use the square of a sum pattern, substituting for and $\sqrt{17}$ for in the pattern:

$(A+ B)^{2}= A^{2}+ 2AB + B^{2}$

$(y + \sqrt{17})^{2}= y^{2}+ 2y \sqrt{17} + (\sqrt{17})^{2}$

$(y + \sqrt{17})^{2}= y^{2}+ 2y \sqrt{17} + 17$

This is not equivalent to any of the given choices.

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Example Question #121 : Exponents

Which of the following is the square of $\frac{1}{4}x + \frac{1}{7}$ ?

Possible Answers:

$\frac{1}{8}x ^{2}+ \frac{1}{28}x +\frac{1}{14}$

$\frac{1}{16}x ^{2}+ \frac{1}{56}x +\frac{1}{49}$

$\frac{1}{16}x ^{2}+ \frac{1}{28}x +\frac{1}{49}$

$\frac{1}{16}x ^{2}+ \frac{1}{14}x +\frac{1}{49}$

$\frac{1}{8}x ^{2}+ \frac{1}{14}x +\frac{1}{14}$

Correct answer:

$\frac{1}{16}x ^{2}+ \frac{1}{14}x +\frac{1}{49}$

Explanation:

Use the square of a sum pattern, substituting $5 \sqrt{x}$ for and $3 \sqrt{y}$ for in the pattern:

$(A+ B)^{2}= A^{2}+ 2AB + B^{2}$

$\left (\frac{1}{4}x + \frac{1}{7} \right )^{2}= \left (\frac{1}{4}x \right )^{2}+ 2 \left (\frac{1}{4}x \right )\left (\frac{1}{7} \right ) + \left (\frac{1}{7} \right )^{2}$

$= \frac{1}{16}x ^{2}+ \frac{1}{14}x +\frac{1}{49}$

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ACT Math : Squaring / Square Roots / Radicals

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1 : How To Find The Square Of Difference

Example Question #1 : How To Find The Square Of Difference

Example Question #1 : How To Find The Square Of Difference

Example Question #113 : Exponents

Example Question #2 : Square Of Sum

Example Question #1 : How To Find The Square Of A Sum

Example Question #2 : Squaring / Square Roots / Radicals

Example Question #4 : Squaring / Square Roots / Radicals

Example Question #6 : How To Find The Square Of A Sum

Example Question #121 : Exponents

All ACT Math Resources

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