To solve the equation \((4x - 5)^2 = 49\), we can start by taking the square root of both sides:
\[ 4x - 5 = 7 \quad \text{or} \quad 4x - 5 = -7 \]
Now, we can solve each case separately.
Case 1:
\[ 4x - 5 = 7 \]
Add 5 to both sides:
\[ 4x = 12 \]
Now, divide by 4:
\[ x = 3 \]
Case 2:
\[ 4x - 5 = -7 \]
Add 5 to both sides:
\[ 4x = -2 \]
Now, divide by 4:
\[ x = -\frac{1}{2} \]
Thus, the possible values of \(x\) are:
\[ x = 3 \quad \text{and} \quad x = -\frac{1}{2} \]
Now let's check if any of the provided options are valid solutions:
- \(-\frac{4}{5}\) -> Not a solution
- \(-\frac{1}{2}\) -> Is a solution
- \(3\) -> Is a solution
- \(5\) -> Not a solution
The possible values of \(x\) that satisfy the equation \((4x - 5)^2 = 49\) are:
\[ -\frac{1}{2} \quad \text{and} \quad 3 \]