How to find the square of a sum - ACT Math

Card 0 of 72

Question

Expand:

Answer

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.

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Question

Which of the following is the square of ?

Answer

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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Question

Evaluate the following expression:

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

Answer

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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Question

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

You may assume both and are positive.

Answer

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

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

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

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

or

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

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Question

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

Answer

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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Question

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

Answer

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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Question

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

Answer

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

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

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

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

Compare your answer with the correct one above

Question

Which of the following is the square of ?

Answer

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

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

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

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

Compare your answer with the correct one above

Question

Expand:

Answer

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.

Compare your answer with the correct one above

Question

Which of the following is the square of ?

Answer

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

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

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

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

Compare your answer with the correct one above

Question

Evaluate the following expression:

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

Answer

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

Compare your answer with the correct one above

Question

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

You may assume both and are positive.

Answer

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

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

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

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

or

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

Compare your answer with the correct one above

Question

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

Answer

Multiply vertically as follows:

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

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

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

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

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

Compare your answer with the correct one above

Question

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

Answer

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.

Compare your answer with the correct one above

Question

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

Answer

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

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

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

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

Compare your answer with the correct one above

Question

Which of the following is the square of ?

Answer

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

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

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

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

Compare your answer with the correct one above

Question

Expand:

Answer

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.

Compare your answer with the correct one above

Question

Which of the following is the square of ?

Answer

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

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

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

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

Compare your answer with the correct one above

Question

Evaluate the following expression:

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

Answer

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

Compare your answer with the correct one above

Question

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

You may assume both and are positive.

Answer

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

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

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

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

or

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

Compare your answer with the correct one above

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