SAT Equivalent Expressions Question 7 | Free SAT Prep | Aniko

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Question 7·Easy·Equivalent Expressions

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Which of the following expressions is equivalent to $x^2 + 6x + 9$ ?

For "equivalent expression" questions with quadratics, first try factoring the given expression: look for two numbers that multiply to the constant term and add to the coefficient of $x$ . If it looks like a perfect square trinomial ( $x^2 + 2ax + a^2$ ), rewrite it as $(x + a)^2$ . As a quick check, you can also expand promising answer choices mentally or plug in a simple value for $x$ (like $x = 0$ or $x = 1$ ) to eliminate options that do not match the original expression.

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Think about factoring the trinomial

Try to factor $x^2 + 6x + 9$ into the form $(x + m)(x + n)$ . What must $m$ and $n$ multiply to, and what must they add to?

Focus on the constant term and the middle coefficient

The constant term is 9, and the coefficient of $x$ is 6. List factor pairs of 9 and check which pair adds up to 6.

Connect your factors to the answer choices

Once you know the two numbers that factor the trinomial, think about how that would look as a binomial squared and compare it to the answer options.

Desmos Guide

Enter the original expression

In Desmos, type f(x) = x^2 + 6x + 9 to define the original function.

Enter each answer choice as a separate function

Type each option as its own function, for example g(x) = (x + 3)^2, h(x) = (x - 3)^2, p(x) = x^2 + 9x + 6, and q(x) = x(x + 6). Then compare their graphs or use a table (click the gear icon and select "Table") to see which one has the same $y$ -values as $f(x)$ for many different $x$ -values; that one is equivalent to $f(x)$ .

Step-by-step Explanation

Recognize the form of the expression

The expression $x^2 + 6x + 9$ is a quadratic trinomial in the standard form $x^2 + bx + c$ . To find an equivalent factored form, we want to factor it into $(x + m)(x + n)$ for some numbers $m$ and $n$ .

Find two numbers that multiply to 9 and add to 6

Look at the constant term $9$ . Its factor pairs are $1$ and $9$ , or $3$ and $3$ . We also need these two numbers to add up to the coefficient of $x$ , which is $6$ . The pair $3$ and $3$ works because $3 \times 3 = 9$ and $3 + 3 = 6$ .

Write the factored form using those numbers

Since the numbers are $3$ and $3$ , the factored form of $x^2 + 6x + 9$ is $(x + 3)(x + 3)$ . This is the same as $(x + 3)^2$ , so $(x + 3)^2$ is the expression equivalent to $x^2 + 6x + 9$ .

Strategy

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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