Question 133·Easy·Equivalent Expressions

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Which expression is equivalent to $4x^{2} - 12x + 9$ ?

Look for perfect-square trinomials by checking whether the first and last terms are squares (here $4x^2=(2x)^2$ and $9=3^2$ ) and whether the middle term matches $-2ab$ (here $-2(2x)(3)=-12x$ ). If all three terms match the pattern, rewrite immediately as $(a-b)^2$ or $(a+b)^2$ rather than expanding every choice.

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Use a standard identity

Try matching $4x^{2}-12x+9$ to $(a-b)^2=a^2-2ab+b^2$ .

Start with the first term

What expression squared gives $4x^2$ ? That will be your $a$ .

Use the middle term to find the constant

After choosing $a$ , use the middle term $-2ab$ to solve for $b$ , then verify that $b^2$ matches the constant term.

Desmos Guide

Enter the original expression

In Desmos, enter $f(x)=4x^2-12x+9$ .

Enter each answer choice as a new function

Enter each option as $g_1(x)=(2x-3)^2$ , $g_2(x)=(2x+3)^2$ , $g_3(x)=(x-3)^2$ , and $g_4(x)=4(x-3)^2$ .

Identify which graph matches exactly

The equivalent expression will produce a graph that lies exactly on top of $f(x)$ for all $x$ . The matching option is $(2x - 3)^{2}$ .

Step-by-step Explanation

Recognize the perfect-square pattern

Compare $4x^{2}-12x+9$ to the identity

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

Here, $4x^2$ suggests $a=2x$ because $(2x)^2=4x^2$ .

Match the middle and last terms

Using $a=2x$ , the middle term of $(a-b)^2$ is $-2ab=-2(2x)b=-4bx$ .

Set $-4bx$ equal to $-12x$ to find $b$ :

-4bx=-12x \Rightarrow b=3.

Check the constant term: $b^2=3^2=9$ , which matches.

Write the equivalent expression

Since $a=2x$ and $b=3$ , the expression factors as

(2x-3)^2.

So the equivalent expression is $(2x - 3)^{2}$ .

Question 133·Easy·Equivalent Expressions

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Desmos Guide

Step-by-step Explanation

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Strategy

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Step-by-step Explanation