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

Question

(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 isolate the squared term.

1. Add 6 to both sides:

\[ (x + 27)^2 - 6 + 6 = -3 + 6 \]

\[ (x + 27)^2 = 3 \]

2. Take the square root of both sides:

\[ x + 27 = \pm \sqrt{3} \]

3. Isolate \(x\):

For the positive square root:

\[ x = -27 + \sqrt{3} \]

For the negative square root:

\[ x = -27 - \sqrt{3} \]

4. Calculate the numerical values of the solutions:

Using the approximation \(\sqrt{3} \approx 1.732\):

For \(x = -27 + \sqrt{3}\):

\[ x \approx -27 + 1.732 \approx -25.268 \]

For \(x = -27 - \sqrt{3}\):

\[ x \approx -27 - 1.732 \approx -28.732 \]

5. Rounding to the nearest hundredth:

• The first solution (the smaller one) rounded is: \[ x \approx -28.73 \]

• The second solution (the larger one) rounded is: \[ x \approx -25.27 \]

6. Listing the answers from smallest to largest:

The first solution is \(x = -28.73\)

The second solution is \(x = -25.27\)

Final answers: The first solution is \(x = -28.73\)

The second solution is \(x = -25.27\)