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

Question

(
𝑥
+
36
)
2
−
38
=
251
.(1 point)
Responses

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

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

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

{53,19}

Answer 1

To solve the quadratic equation \((x + 36)^2 - 38 = 251\), we will first isolate the squared term. Here's a step-by-step solution:

1. Start by moving \(-38\) to the right side of the equation:

   \[ (x + 36)^2 = 251 + 38 \]

   This simplifies to:

   \[ (x + 36)^2 = 289 \]

2. Next, take the square root of both sides:

   \[ x + 36 = \pm \sqrt{289} \]

   Since the square root of \(289\) is \(17\):

   \[ x + 36 = 17 \quad \text{or} \quad x + 36 = -17 \]

3. Now, solve for \(x\) in both cases:

   For \(x + 36 = 17\):

   \[ x = 17 - 36 = -19 \]

   For \(x + 36 = -17\):

   \[ x = -17 - 36 = -53 \]

4. Therefore, the solutions to the equation are:

   \[ x = -19 \quad \text{and} \quad x = -53 \]

So, the solution set is \({-19, -53}\). The correct response from the options given is:

\({−19,−53}\)