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

Question

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) \({-19, -53}\).