Solve the following quadratic equation using square roots. Round to the nearest hundredth if necessary: (x − 6)^2 = 51.

Answer 1

To solve the quadratic equation \((x - 6)^2 = 51\) using square roots, we first take the square root of both sides of the equation.

1. Start by isolating the square term: \[ (x - 6)^2 = 51 \]

2. Take the square root of both sides: \[ x - 6 = \pm \sqrt{51} \]

   This gives us two equations to solve: \[ x - 6 = \sqrt{51} \] and \[ x - 6 = -\sqrt{51} \]

3. Solve for \(x\) in both cases:

   For the first case: \[ x = 6 + \sqrt{51} \]

   For the second case: \[ x = 6 - \sqrt{51} \]

4. Next, we calculate \(\sqrt{51}\):

   • The approximate value of \(\sqrt{51}\) is about \(7.14\) (since \(7.14^2 \approx 51\)).

5. Now substitute this value back into the equations:

   For \(x = 6 + \sqrt{51}\): \[ x \approx 6 + 7.14 = 13.14 \]

   For \(x = 6 - \sqrt{51}\): \[ x \approx 6 - 7.14 = -1.14 \]

6. Thus, the solutions to the equation \((x - 6)^2 = 51\) are approximately: \[ x \approx 13.14 \quad \text{and} \quad x \approx -1.14 \]

Finally, rounding to the nearest hundredth, the solutions are: \[ x \approx 13.14 \quad \text{and} \quad x \approx -1.14 \]