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ACT Math : Algebra

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Example Questions

Example Question #2 : How To Find The Square Of A Sum

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 #1 : How To Find The Square Of A Sum

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

Possible Answers:

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

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

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

The correct answer is not given among the other responses.

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

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 #2 : Squaring / Square Roots / Radicals

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

Possible Answers:

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

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

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

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

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

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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Example Question #6 : Square Of Sum

Which of the following is the square of 3.7x + 1.4 y ?

Possible Answers:

$13.69x^{2}+ 10.36x y +1.96 y ^{2}$

$13.69x^{2}+ 5.18x y +1.96 y ^{2}$

$7.4x^{2}+ 10.36x y +2.8 y ^{2}$

$13.69x^{2} +1.96 y ^{2}$

$7.4x^{2}+ 5.18x y +2.8 y ^{2}$

Correct answer:

$13.69x^{2}+ 10.36x y +1.96 y ^{2}$

Explanation:

Use the square of a sum pattern, substituting 3.7x for and 1.4 y for in the pattern:

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

$\left (3.7x + 1.4 y \right )^{2}= (3.7x)^{2}+ 2 (3.7x)(1.4 y) + (1.4 y) ^{2}$

$= 13.69x^{2}+ 10.36x y +1.96 y ^{2}$

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Example Question #821 : Algebra

Which real number satisfies $3^{x}*9=27^{2}$ ?

Possible Answers:

Correct answer:

Explanation:

Simplify the base of 9 and 27 in order to have a common base.

(3^x)(9)=27²

= (3^x)(3²)=(3³)²

=(3^x+2)=3⁶

Therefore:

x+2=6

x=4

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Example Question #2 : Factoring Squares

Which of the following is a factor of $4x^{4}+ 36 x^{2}$ ?

Possible Answers:

$2x^{2}+ 3$

$2x^{2}+ 9$

$x^{2}+ 9$

$x^{2}+ 6$

$x^{2}+ 3$

Correct answer:

$x^{2}+ 9$

Explanation:

The terms of $4x^{4}+ 36 x^{2}$ have $4x^{2}$ as their greatest common factor, so

$4x^{4}+ 36 x^{2} = 4x^{2} (x^{2}+9)$

$x^{2}+ 9$ is a prime polynomial.

Of the five choices, only $x^{2}+ 9$ is a factor.

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Example Question #822 : Algebra

Simplify $\frac{8^{3} \times 3}{2^{6}\times 27}$

Possible Answers:

$\frac{8}{9}$

$\frac{4}{3}$

$\frac{16}{9}$

$\frac{9}{8}$

$\frac{32}{9}$

Correct answer:

$\frac{8}{9}$

Explanation:

The easiest way to approach this problem is to break everything into exponents. $8^{3}$ is equal to $2^{9}$ and 27 is equal to $3^{3}$ . Therefore, the expression can be broken down into $\frac{2^{9} \times 3}{2^{6}\times 3^{3}}$ . When you cancel out all the terms, you get $\frac{2^{3}}{3^{2}}$ , which equals $\frac{8}{9}$ .

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Example Question #2 : Squaring / Square Roots / Radicals

Which of the following expression is equal to

$\sqrt{(45)(9)+(27)(36)}$

Possible Answers:

$7\sqrt{17}$

$9\sqrt{19}$

$9\sqrt{17}$

$9\sqrt{15}$

$7\sqrt{15}$

Correct answer:

$9\sqrt{17}$

Explanation:

$\sqrt{(45)(9)+(27)(36)}$

When simplifying a square root, consider the factors of each of its component parts:

$\sqrt{(3^2)\times5\times(3^{2})+(3^{3})(2^2)(3^2)}$

Combine like terms:

$\sqrt{5(3^4)+(2^2)(3^5)}$

Remove the common factor, 3^4 :

$\sqrt{(3^4)\times(5+3\times(2^2))}$

Pull the $\sqrt{3^4}$ outside of the equation as 3^2 :

$(3^2)\sqrt{5+12}=9\sqrt{17}$

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Example Question #3 : Squaring / Square Roots / Radicals

Which of the following is equal to the following expression?

$\sqrt{(16)(8)+(32)(20)}$

Possible Answers:

$2^{3}\sqrt{10}$

$2^{7}\sqrt{5}$

$2^{3}\sqrt{6}$

$2^{4}\sqrt{5}$

$2^4\sqrt{3}$

Correct answer:

$2^4\sqrt{3}$

Explanation:

$\sqrt{(16)(8)+(32)(20)}$

First, break down the components of the square root:

$\sqrt{(2^{4})(2^{3})+(2^{5})(2^{2})\times 5}$

Combine like terms. Remember, when multiplying exponents, add them together:

$\sqrt{(2^{7})+(2^{7})\times5}$

Factor out the common factor of 2^7 :

$\sqrt{(2^{7})(1+5)}$

$\sqrt{(2^7)\times6}$

Factor the :

$\sqrt{(2^7)\times2\times3}$

Combine the factored with the 2^7 :

$\sqrt{(2^{8})\times3}$

Now, you can pull $\sqrt{2^8}$ out from underneath the square root sign as 2^4 :

$2^4\sqrt{3}$

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ACT Math : Algebra

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #2 : How To Find The Square Of A Sum

Example Question #4 : Squaring / Square Roots / Radicals

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

Example Question #2 : Squaring / Square Roots / Radicals

Example Question #6 : Square Of Sum

Example Question #821 : Algebra

Example Question #2 : Factoring Squares

Example Question #822 : Algebra

Example Question #2 : Squaring / Square Roots / Radicals

Example Question #3 : Squaring / Square Roots / Radicals

All ACT Math Resources

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