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 can begin by isolating the squared term. Here are the steps to solve it:

1. **Add 38 to both sides of the equation**:
\[
(x + 36)^2 - 38 + 38 = 251 + 38
\]
\[
(x + 36)^2 = 289
\]

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:
- For the first case:
\[
x + 36 = 17
\]
\[
x = 17 - 36
\]
\[
x = -19
\]

- For the second case:
\[
x + 36 = -17
\]
\[
x = -17 - 36
\]
\[
x = -53
\]

4. **Final answers**:
The two solutions are:
\[
x = -19 \quad \text{and} \quad x = -53
\]

Thus, the solutions to the equation are \(x = -19\) and \(x = -53\).