Question

Solve the following quadratic equation using square roots: (x + 36)^2 − 38 = 251.
A) {53, 19}
B) {253, -325}
C) {17.84, -13.84}
D) {-19, -53}

Answer 1

To solve the quadratic equation \((x + 36)^2 - 38 = 251\) using square roots, follow these steps:

1. Start by isolating the squared term:
\[
(x + 36)^2 - 38 = 251
\]
Add 38 to both sides:
\[
(x + 36)^2 = 251 + 38
\]
\[
(x + 36)^2 = 289
\]

2. Take the square root of both sides:
\[
x + 36 = \pm\sqrt{289}
\]
Since \(\sqrt{289} = 17\), we have:
\[
x + 36 = 17 \quad \text{or} \quad x + 36 = -17
\]

3. Solve for \(x\) in both cases:

**Case 1:**
\[
x + 36 = 17
\]
Subtract 36 from both sides:
\[
x = 17 - 36 = -19
\]

**Case 2:**
\[
x + 36 = -17
\]
Subtract 36 from both sides:
\[
x = -17 - 36 = -53
\]

4. The solutions to the equation are:
\[
x = -19 \quad \text{and} \quad x = -53
\]

Thus, the final answer is:
\[
\{-19, -53\}
\]

The correct option is D) \(\{-19, -53\}\).