Solve the following quadratic equation using square roots: (x+36) to the second power − 38=251

Question

.(1 point)
Responses

{53,19}
left brace 53 comma 19 right brace

{−19,−53}
left brace negative 19 comma negative 53 right brace

{253,−325}
left brace 253 comma negative 325 right brace

{17.84,−13.84}
left brace 17.84 comma negative 13.84 right brace

Answer 1

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

1. Isolate 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 \(\sqrt{289} = 17\): \[ x + 36 = \pm 17 \]

3. Solve for x by considering both the positive and negative square roots:

   • For the positive case: \[ x + 36 = 17 \] Subtract 36 from both sides: \[ x = 17 - 36 = -19 \]

   • For the negative case: \[ x + 36 = -17 \] Subtract 36 from both sides: \[ x = -17 - 36 = -53 \]

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

Therefore, the response is \({-19, -53}\).