Question

solve the following quadratic equation using square roots 19+(x-3)^2/-4 = -4

{2, -8}
{8, -2}
{28, -22}
{16, -10}

Answer 1

solve the following quadratic equation using square roots 19+(x-3)^2/-4 = -4

A. {2, -8}
B. {8, -2}
C. {28, -22}
D. {16, -10}

Answer 2

To solve the quadratic equation \( 19 + \frac{(x-3)^2}{-4} = -4 \) using square roots, we will follow these steps:

1. Isolate the square term.
2. Eliminate the fraction by multiplying through by \(-4\).
3. Solve for \(x\).

### Step 1: Isolate the square term
Start with the original equation:
\[
19 + \frac{(x-3)^2}{-4} = -4
\]

Subtract 19 from both sides:
\[
\frac{(x-3)^2}{-4} = -4 - 19
\]
\[
\frac{(x-3)^2}{-4} = -23
\]

Now, multiply both sides by \(-4\):
\[
(x-3)^2 = 92
\]

### Step 2: Solve for \(x\)
Now take the square root of both sides:
\[
x - 3 = \pm \sqrt{92}
\]

This can be simplified:
\[
x - 3 = \pm 2\sqrt{23}
\]

Now, solve for \(x\):
\[
x = 3 \pm 2\sqrt{23}
\]

### Step 3: Numerical approximations
Calculate the two possible values of \(x\):
1. \( x = 3 + 2\sqrt{23} \)
2. \( x = 3 - 2\sqrt{23} \)

Calculating \(2\sqrt{23} \approx 2 \times 4.796 = 9.592\):
- For \(x = 3 + 2\sqrt{23}\):
\[
x \approx 3 + 9.592 \approx 12.592 \quad (\text{approximately } 12.59)
\]

- For \(x = 3 - 2\sqrt{23}\):
\[
x \approx 3 - 9.592 \approx -6.592 \quad (\text{approximately } -6.59)
\]

### Final Answer
The approximate solutions do not match any of the provided pairs directly. However, you can summarize the two roots as:
- \(x \approx 12.59\)
- \(x \approx -6.59\)

Since our numerical approximations don't directly correspond to any items listed, this may suggest the options given were from a different question or not approximated directly.

The solutions in terms of forms \(3 \pm 2\sqrt{23}\) may not equate to the pairs provided in your multiple choices. If you need exact decimal values computed or further manipulation, please clarify!

Answer 3

To solve the quadratic equation \( 19 + \frac{(x-3)^2}{-4} = -4 \) using square roots, let's go through the steps systematically.

### Step 1: Isolate the Square Term

Starting with the equation:

\[
19 + \frac{(x-3)^2}{-4} = -4
\]

Subtract 19 from both sides:

\[
\frac{(x-3)^2}{-4} = -4 - 19
\]

This simplifies to:

\[
\frac{(x-3)^2}{-4} = -23
\]

### Step 2: Multiply by -4

Now, multiply both sides by \(-4\) to eliminate the fraction:

\[
(x-3)^2 = 92
\]

### Step 3: Take the Square Root

Now take the square root of both sides:

\[
x - 3 = \pm \sqrt{92}
\]

Recall that \( \sqrt{92} = \sqrt{4 \cdot 23} = 2\sqrt{23} \):

\[
x - 3 = \pm 2\sqrt{23}
\]

### Step 4: Solve for x

Now solve for \(x\):

\[
x = 3 + 2\sqrt{23} \quad \text{or} \quad x = 3 - 2\sqrt{23}
\]

### Step 5: Calculate Numerical Values

Calculating \(2\sqrt{23}\):

1. Approximately, \( \sqrt{23} \approx 4.796 \).
2. Thus, \( 2\sqrt{23} \approx 9.592 \).

Calculating the solutions:

- For \( x = 3 + 2\sqrt{23} \):
\[
x \approx 3 + 9.592 \approx 12.592
\]

- For \( x = 3 - 2\sqrt{23} \):
\[
x \approx 3 - 9.592 \approx -6.592
\]

### Step 6: Check Options

We need to determine if these values are related to any of the answer choices:

**A. {2, -8}**
**B. {8, -2}**
**C. {28, -22}**
**D. {16, -10}**

Since none of the approximate values \( \{12.592, -6.592\} \) match any of the provided options, we should also check if we made an error in our calculations or if the problem might be different from what’s presented.

### Conclusion

Given the computations, neither \( \{12.592, -6.592\} \) matches any of the answers. Please reconfirm the problem statement or the options given. Would you like me to assist you further with a different aspect of the problem or with similar problems?