Consider the expression The expression can be rewritten as , where , , and are constants. Which choice is equal to ?

Solution available with step-by-step explanation

Super-Hard SAT Math Question 16 | Free SAT Prep | Aniko

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Question 16·200 Super-Hard SAT Math Questions·Advanced Math

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Consider the expression

11(4x-3)^2-7(4x+1)^2+6(4x-3)(4x+1)

The expression can be rewritten as $\frac{a}{4}x^2+\frac{b}{4}x+\frac{c}{4}$ , where $a$ , $b$ , and $c$ are constants. Which choice is equal to $a+b+c$ ?

When an expression is rewritten in a form like $\frac{a}{k}x^2+\frac{b}{k}x+\frac{c}{k}$ and you need $a+b+c$ , avoid expanding. Instead, plug in $x=1$ to turn $x^2$ , $x$ , and the constant into 1, giving $E(1)=\frac{a+b+c}{k}$ . Then multiply by $k$ . This approach is faster and reduces algebra errors compared with full expansion.

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Use a smart input value

Because $a+b+c$ is a sum of coefficients, try plugging in a value of $x$ that makes $x^2$ , $x$ , and the constant all become 1.

Translate the new form into a quick computation

If the expression equals $\frac{a}{4}x^2+\frac{b}{4}x+\frac{c}{4}$ , what does the expression become when $x=1$ ?

Evaluate the original expression efficiently

At $x=1$ , compute $4x-3$ and $4x+1$ first, then substitute those numbers into the expression before doing any expansion.

Desmos Guide

Define the expression as a function

Enter

f(x)=11(4x-3)^2-7(4x+1)^2+6(4x-3)(4x+1)

as a function.

Evaluate the function at $x=1$

In a new line, type f(1) and note the value Desmos returns.

Scale to get $a+b+c$

Because $f(1)=\frac{a+b+c}{4}$ , type 4f(1) and match the result to the answer choices. This value is $a+b+c$ .

Step-by-step Explanation

Connect $a+b+c$ to the value at $x=1$

Let

E(x)=\frac{a}{4}x^2+\frac{b}{4}x+\frac{c}{4}.

Substitute $x=1$ :

E(1)=\frac{a}{4}+\frac{b}{4}+\frac{c}{4}=\frac{a+b+c}{4}.

So $a+b+c=4E(1)$ . This means you can find $a+b+c$ without fully expanding the whole expression.

Evaluate the original expression at $x=1$

When $x=1$ :

$4x-3=4(1)-3=1$
$4x+1=4(1)+1=5$

Substitute into the given expression:

E(1)=11(1)^2-7(5)^2+6(1)(5).

Compute:

E(1)=11-7\cdot 25+30=11-175+30=-134.

Multiply by 4 to get $a+b+c$

Since $a+b+c=4E(1)$ ,

a+b+c=4(-134)=-536.

Therefore, the correct choice is -536.

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Step-by-step Explanation

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Step-by-step Explanation