Solve the following quadratic equation using square roots. Round to the nearest hundredth if necessary: (x+27)2−6=−3

Answer 1

To solve the quadratic equation \((x + 27)^2 - 6 = -3\), we start by isolating the square term.

1. Add 6 to both sides to eliminate the \(-6\): \[ (x + 27)^2 - 6 + 6 = -3 + 6 \] This simplifies to: \[ (x + 27)^2 = 3 \]

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

3. Solve for \(x\) by isolating it: \[ x = -27 \pm \sqrt{3} \]

This gives us two potential solutions:

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

4. Calculate the numerical values:

   • For \(x = -27 + \sqrt{3}\): \[ \sqrt{3} \approx 1.732 \] \[ x \approx -27 + 1.732 \approx -25.268 \]
   • For \(x = -27 - \sqrt{3}\): \[ x \approx -27 - 1.732 \approx -28.732 \]

5. Round the results to the nearest hundredth:

   • First solution: \[ x \approx -25.27 \]
   • Second solution: \[ x \approx -28.73 \]

Thus, the solutions to the quadratic equation are: \[ x \approx -25.27 \quad \text{and} \quad x \approx -28.73 \]