The expression above is equivalent to , where , , and are constants. What is the value of ?

Solution available with step-by-step explanation

SAT Equivalent Expressions Question 95 | Free SAT Prep | Aniko

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Question 95·Medium·Equivalent Expressions

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(3x-5)^2-(x+1)(x-1)

The expression above is equivalent to $ax^2+bx+c$ , where $a$ , $b$ , and $c$ are constants. What is the value of $b$ ?

Your answer

(Express the answer as an integer)

For questions asking for a specific coefficient after rewriting an expression, first expand all products using FOIL or known formulas like $(a-b)^2=a^2-2ab+b^2$ and $(a+b)(a-b)=a^2-b^2$ . Then carefully handle any minus signs in front of parentheses, distribute them, and combine like terms in order: $x^2$ terms, $x$ terms, then constants. Finally, read the requested coefficient directly from the simplified expression, double-checking sign errors since those are the most common source of mistakes.

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Hint 1: Expand each product

Before trying to match the form $ax^2+bx+c$ , rewrite both $(3x-5)^2$ and $(x+1)(x-1)$ as standard polynomials by expanding them.

Hint 2: Be careful with the minus sign

After you expand $(x+1)(x-1)$ , remember that the entire expression is being subtracted; distribute the negative sign to both terms inside the parentheses.

Hint 3: Identify the coefficient of $x$

Once you have a single quadratic expression, group like terms. The value of $b$ is whatever number multiplies $x$ in the final simplified expression.

Desmos Guide

Enter the given expression

In Desmos, type y = (3x - 5)^2 - (x + 1)(x - 1) to graph the quadratic expression.

Create a general quadratic for comparison

On a new line, type y = a x^2 + b x + c and let Desmos create sliders for a, b, and c.

Match the graphs to read off $b$

Adjust the sliders for a, b, and c until the graph of y = a x^2 + b x + c lies exactly on top of the graph of y = (3x - 5)^2 - (x + 1)(x - 1). The value shown for the b slider is the coefficient of $x$ in the equivalent quadratic.

Step-by-step Explanation

Expand $(3x-5)^2$

Use FOIL or the square-of-a-binomial formula $(a-b)^2=a^2-2ab+b^2$ .

Here, $a=3x$ and $b=5$ :

(3x-5)^2=(3x)^2-2(3x)(5)+5^2 =9x^2-30x+25

Expand $(x+1)(x-1)$

Recognize this as a difference of squares: $(x+1)(x-1)=x^2-1^2$ .

So:

(x+1)(x-1)=x^2-1

Subtract the second expression from the first

The original expression is

(3x-5)^2-(x+1)(x-1)

Substitute the expanded forms from Steps 1 and 2:

9x^2-30x+25 - (x^2-1)

Now distribute the minus sign through the parentheses:

9x^2-30x+25 - x^2 + 1

Combine like terms and identify $b$

Combine like terms:

$x^2$ terms: $9x^2 - x^2 = 8x^2$
$x$ terms: $-30x$ stays as is
Constant terms: $25 + 1 = 26$

So the expression becomes

8x^2 - 30x + 26

This matches the form $ax^2+bx+c$ , so $b$ is the coefficient of $x$ , which is $-30$ .

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