SAT Equivalent Expressions Question 175 | Free SAT Prep | Aniko

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Question 175·Easy·Equivalent Expressions

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Which expression is equivalent to $13(u^2 - 49v^2)$ ?

When you see something like $x^2 - k$ or, more generally, $a^2 - b^2$ , immediately check if both terms are perfect squares so you can use the difference of squares pattern $a^2 - b^2 = (a - b)(a + b)$ . Keep any coefficients outside the parentheses (like the 13 here) as they are, factor just the inside using the pattern, and then quickly eliminate wrong choices by expanding in your head or checking that the squared term and the constant match the original.

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Look inside the parentheses first

Focus on $u^2 - 49v^2$ . Can you rewrite $49v^2$ as a square of something simpler?

Recognize a factoring pattern

Once you write $49v^2$ as $(7v)^2$ , what well-known pattern does $u^2 - (7v)^2$ look like?

Match the pattern carefully

Use $a^2 - b^2 = (a - b)(a + b)$ with the correct choices of $a$ and $b$ , and then make sure the 13 in front is still there.

Desmos Guide

Enter the original expression

In one expression line, type 13(u^2 - 49v^2). Desmos will create sliders for u and v so you can assign them values.

Enter each answer choice as a separate expression

On new lines, type:

13(u - 7v)(u + 7v)
13(u - 49v)(u + v)
13(u - 7v)(u - 7v)
13(u - 14v)(u + 14v) Desmos will show a numeric value for each once you set values for u and v.

Test several values of u and v

Move the sliders or type in values for u and v (for example, u = 2, v = 3, then u = -1, v = 4). For each pair of values, compare the numbers Desmos gives for the original expression and for each option. The expression that always matches the original for every tested pair is the equivalent one.

Step-by-step Explanation

Identify the structure inside the parentheses

Look at the part inside the 13: $u^2 - 49v^2$ .

Notice that $u^2$ is a perfect square and $49v^2$ is also a perfect square, because $49v^2 = (7v)^2$ . So we can rewrite the expression as:

13(u^2 - (7v)^2)

This is a difference of squares.

Recall the difference of squares formula

The standard factoring pattern for a difference of squares is:

a^2 - b^2 = (a - b)(a + b)

Here, compare $u^2 - (7v)^2$ with $a^2 - b^2$ :

$a$ corresponds to $u$ .
$b$ corresponds to $7v$ .

Apply the formula and include the 13

Using $a = u$ and $b = 7v$ in the formula $a^2 - b^2 = (a - b)(a + b)$ , we factor:

u^2 - (7v)^2 = (u - 7v)(u + 7v)

Now put back the 13 that was in front:

13(u^2 - 49v^2) = 13(u - 7v)(u + 7v)

So the equivalent expression is $13(u - 7v)(u + 7v)$ .

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation