Let's solve the quadratic equation step by step.
The given equation is:
\[ -9 + \frac{(x - 3)^2}{4} = -4 \]
First, we want to isolate the term involving \(x\). To do this, we can add 9 to both sides of the equation:
\[ \frac{(x - 3)^2}{4} = -4 + 9 \]
Calculating the right side gives:
\[ \frac{(x - 3)^2}{4} = 5 \]
Next, we can eliminate the fraction by multiplying both sides of the equation by 4:
\[ (x - 3)^2 = 20 \]
Now, we will take the square root of both sides. Remember, taking the square root will yield both a positive and a negative solution:
\[ x - 3 = \pm \sqrt{20} \]
We can simplify \(\sqrt{20}\):
\[ \sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5} \]
Therefore, we have:
\[ x - 3 = \pm 2\sqrt{5} \]
Next, we can solve for \(x\):
-
Using the positive root: \[ x = 3 + 2\sqrt{5} \]
-
Using the negative root: \[ x = 3 - 2\sqrt{5} \]
Now, let's approximate the values of \(2\sqrt{5}\):
\[ \sqrt{5} \approx 2.236 \implies 2\sqrt{5} \approx 4.472 \]
Thus, we have:
- \(x \approx 3 + 4.472 \approx 7.472\)
- \(x \approx 3 - 4.472 \approx -1.472\)
So, the solutions are approximately \(7.472\) and \(-1.472\).
Comparing these with the provided options:
- {8, -2}
- {2, -8}
- {16, -10}
- {28, - 22}
It seems none of the provided answer options accurately reflect the solutions we calculated. Therefore, the solutions of the given quadratic equation do not match any options provided.
The approximate solutions are \(7.472\) and \(-1.472\). If these were intended to be rounded values of the options you have, please check them again.