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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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1 answer

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:

  1. Calculate the two solutions:

    • First solution (smallest): \[ x = -27 - \sqrt{3} \]
    • Second solution (largest): \[ x = -27 + \sqrt{3} \]

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

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

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

  5. 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 \(x = -25.27\)