For the quadratic equation (x − 7)^2 = 11, complete the table by testing each of the given values to determine whether it is a solution. Identify the two solutions to the quadratic equation.

Question

The smaller of the two solutions is x = ___. The larger of the two solutions is x = ___.

Answer 1

To solve the quadratic equation \((x - 7)^2 = 11\), we first take the square root of both sides:

\[ x - 7 = \pm \sqrt{11} \]

This gives us two equations to solve:

1. \(x - 7 = \sqrt{11}\)
2. \(x - 7 = -\sqrt{11}\)

Now we can solve each equation for \(x\):

1. For \(x - 7 = \sqrt{11}\): \[ x = 7 + \sqrt{11} \]

2. For \(x - 7 = -\sqrt{11}\): \[ x = 7 - \sqrt{11} \]

Next, let's evaluate the approximate values of the solutions. Since \(\sqrt{11} \approx 3.3166\):

1. \(x = 7 + \sqrt{11} \approx 7 + 3.3166 \approx 10.3166\)
2. \(x = 7 - \sqrt{11} \approx 7 - 3.3166 \approx 3.6834\)

Now we identify the smaller and larger solutions:

• The smaller of the two solutions is \(x = 7 - \sqrt{11}\).
• The larger of the two solutions is \(x = 7 + \sqrt{11}\).

Therefore, we finalize our answers:

The smaller of the two solutions is \(x = 7 - \sqrt{11} \approx 3.6834\).
The larger of the two solutions is \(x = 7 + \sqrt{11} \approx 10.3166\).

To fill in the blanks:

The smaller of the two solutions is \(x = 7 - \sqrt{11}\).
The larger of the two solutions is \(x = 7 + \sqrt{11}\).