Question

9.

Solve the following quadratic equation using square roots. Round to the nearest hundredth if necessary. List the answer from smallest to largest number.

(x + 27)2−6 = −3

Here is a number line to help you remember the order of numbers.



(1 point)
The first solution is x=

The second solution is x=

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Answer 1

To solve the quadratic equation \((x + 27)^2 - 6 = -3\), we will first simplify it step by step.

1. **Isolate the squared term**:
\[
(x + 27)^2 - 6 = -3
\]
Adding 6 to both sides:
\[
(x + 27)^2 = 3
\]

2. **Take the square root of both sides**:
\[
x + 27 = \pm \sqrt{3}
\]

3. **Isolate \(x\)**:
\[
x = -27 \pm \sqrt{3}
\]

Now we can find the two solutions:

4. **Calculate the two solutions**:
- **First solution (smallest)**:
\[
x = -27 - \sqrt{3}
\]
- **Second solution (largest)**:
\[
x = -27 + \sqrt{3}
\]

5. **Use a calculator to find \(\sqrt{3}\)**:
\[
\sqrt{3} \approx 1.732
\]

6. **Calculate the first solution**:
\[
x_1 = -27 - 1.732 \approx -28.732
\]

7. **Calculate the second solution**:
\[
x_2 = -27 + 1.732 \approx -25.268
\]

8. **Round to the nearest hundredth**:
- First solution: \(x_1 \approx -28.73\)
- Second solution: \(x_2 \approx -25.27\)

Now, listing the solutions from smallest to largest:
- The first solution is \(x \approx -28.73\).
- The second solution is \(x \approx -25.27\).

**Final Answers**:
- The first solution is \(x = -28.73\)
- The second solution is \(x = -25.27\)