Question

Solve the following quadratic equation using square roots: (x + 36)^2 - 38 = 251

Answer 1

To solve the quadratic equation \((x + 36)^2 - 38 = 251\), we start by isolating the square term.

1. Add 38 to both sides:
\[
(x + 36)^2 = 251 + 38
\]
Simplifying the right side gives:
\[
(x + 36)^2 = 289
\]

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

3. Now, we split this into two cases:
- Case 1: \(x + 36 = 17\)
- Case 2: \(x + 36 = -17\)

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

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

6. Thus, the two solutions to the equation are:
\[
x = -19 \quad \text{and} \quad x = -53
\]

Therefore, the solutions are:
\[
\boxed{-19} \quad \text{and} \quad \boxed{-53}
\]