Question

solve for x in the equation $x^2 + 20x + 100 = 36$.
$\bigcirc$ $x = -16$ or $x = -4$
$\bigcirc$ $x = -10$
$\bigcirc$ $x = -8$
$\bigcirc$ $x = 4$ or $x = 16$

Explanation:

Step1: Factor the left side

The left side \(x^2 + 20x + 100\) is a perfect square trinomial, which factors as \((x + 10)^2\). So the equation becomes \((x + 10)^2 = 36\).

Step2: Take square roots

Take the square root of both sides: \(x + 10=\pm\sqrt{36}\). Since \(\sqrt{36} = 6\), we have \(x + 10=\pm6\).

Step3: Solve for x (case 1)

For the positive case: \(x + 10 = 6\). Subtract 10 from both sides: \(x=6 - 10=-4\).

Step4: Solve for x (case 2)

For the negative case: \(x + 10=-6\). Subtract 10 from both sides: \(x=-6 - 10=-16\).

Answer:

\(x = -16\) or \(x = -4\) (corresponding to the first option: \(x = -16\) or \(x = -4\))