To solve the quadratic equation \((X + 36)^2 - 38 = 251\) using square roots, we first need to isolate the squared term. Here are the steps to solve the equation:
-
Start with the original equation: \[ (X + 36)^2 - 38 = 251 \]
-
Add 38 to both sides to isolate the squared term: \[ (X + 36)^2 = 251 + 38 \] \[ (X + 36)^2 = 289 \]
-
Now, take the square root of both sides: \[ X + 36 = \pm \sqrt{289} \]
-
Since \(\sqrt{289} = 17\), we have: \[ X + 36 = 17 \quad \text{or} \quad X + 36 = -17 \]
-
Now, solve for \(X\) in both cases:
- For \(X + 36 = 17\): \[ X = 17 - 36 = -19 \]
- For \(X + 36 = -17\): \[ X = -17 - 36 = -53 \]
-
Thus, the solutions to the quadratic equation are: \[ X = -19 \quad \text{and} \quad X = -53 \]
The final answer is: \[ X = -19 \quad \text{or} \quad X = -53 \]