Question

select the correct answer.
what is the factored form of this expression?
$x^2 - 12x + 36$
a. $(x - 6)^2$
b. $(x - 6)(x + 6)$
c. $(x + 6)^2$
d. $(x - 12)(x - 3)$

Explanation:

Step1: Recall perfect square formula

The perfect square trinomial formula is \(a^2 - 2ab + b^2=(a - b)^2\).

Step2: Identify \(a\) and \(b\) in the expression

For the expression \(x^2-12x + 36\), we have \(a = x\) (since \(x^2=a^2\)), and \(2ab = 12x\). Substituting \(a=x\) into \(2ab = 12x\), we get \(2\times x\times b=12x\), which simplifies to \(2b = 12\), so \(b = 6\). Also, \(b^2=6^2 = 36\), which matches the constant term in the expression.

Step3: Apply the perfect square formula

Using the formula \(a^2-2ab + b^2=(a - b)^2\) with \(a = x\) and \(b = 6\), we get \(x^2-12x + 36=(x - 6)^2\).

We can also check other options:

• Option B: \((x - 6)(x + 6)=x^2-36\), which is not equal to \(x^2-12x + 36\).
• Option C: \((x + 6)^2=x^2+12x + 36\), which is not equal to \(x^2-12x + 36\).
• Option D: \((x - 12)(x - 3)=x^2-3x-12x + 36=x^2-15x + 36\), which is not equal to \(x^2-12x + 36\).

Answer:

A. \((x - 6)^2\)