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: \[ \boxed{-19} \quad \text{and} \quad \boxed{-53} \]