Question

guided example: factor each expression.
a) ( x^2 + 12x + 36 )
b) ( x^2 - 12x + 36 )

Explanation:

Step1: Factor \(x^2 + 12x + 36\)

Notice it's a perfect square trinomial. The formula for \((a + b)^2 = a^2 + 2ab + b^2\). Here, \(a = x\), \(2ab = 12x\) (so \(b = 6\) since \(2\times x\times6 = 12x\)), and \(b^2 = 36\). So \(x^2 + 12x + 36=(x + 6)^2\).

Step2: Factor \(x^2 - 12x + 36\)

This is also a perfect square trinomial. The formula for \((a - b)^2 = a^2 - 2ab + b^2\). Here, \(a = x\), \(2ab = 12x\) (so \(b = 6\) since \(2\times x\times6 = 12x\)), and \(b^2 = 36\). So \(x^2 - 12x + 36=(x - 6)^2\).

Answer:

a) \((x + 6)^2\)
b) \((x - 6)^2\)