How to find the square of a sum

Help Questions

ACT Math › How to find the square of a sum

Questions 1 - 8

1

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

You may assume both and are positive.

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

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

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

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

Explanation

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

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

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

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

or

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

2

Which of the following is the square of ?

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

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

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

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

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

Explanation

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

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

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

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

3

Which of the following is the square of ?

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

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

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

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

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

Explanation

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

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

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

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

4

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

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

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

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

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

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

Explanation

Multiply vertically as follows:

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

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

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

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

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

5

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

The correct answer is not given among the other responses.

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

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

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

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

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.

6

Expand:

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.

7

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

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

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

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

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

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

Explanation

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

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

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

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

8

Evaluate the following expression:

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

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

Return to subject