Question

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

Explanation:

Step1: Recall the perfect square trinomial formula

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

Step2: Identify \(a\) and \(b\) in the given 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 \(b = 6\) (dividing both sides by \(2x\)). Also, \(b^2=6^2 = 36\), which matches the constant term in the expression.

Step3: Apply the perfect square trinomial formula

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

Answer:

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