Question

which expression is equivalent to $x^2 + 9x - 36$
a. $(x - 6)^2$
b. $(x + 6)(x - 6)$
c. $(x - 9)(x + 4)$
d. $(x - 3)(x + 12)$

Explanation:

Step1: Expand Option A

Expand \((x - 6)^2\) using \((a - b)^2 = a^2 - 2ab + b^2\). Here, \(a = x\), \(b = 6\), so \((x - 6)^2 = x^2 - 12x + 36\). Not equivalent to \(x^2 + 9x - 36\).

Step2: Expand Option B

Expand \((x + 6)(x - 6)\) using \((a + b)(a - b)=a^2 - b^2\). So \((x + 6)(x - 6)=x^2 - 36\). Not equivalent to \(x^2 + 9x - 36\).

Step3: Expand Option C

Expand \((x - 9)(x + 4)\) using distributive property: \(x(x + 4)-9(x + 4)=x^2 + 4x - 9x - 36=x^2 - 5x - 36\). Not equivalent to \(x^2 + 9x - 36\).

Step4: Expand Option D

Expand \((x - 3)(x + 12)\) using distributive property: \(x(x + 12)-3(x + 12)=x^2 + 12x - 3x - 36=x^2 + 9x - 36\). Equivalent to the given expression.

Answer:

D. \((x - 3)(x + 12)\)