For the quadratic equation (x−7)2=11

To solve the quadratic equation \((x - 7)^2 = 11\), we start by taking 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 solve each equation for \(x\):

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

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

Now we can identify the two solutions:

• The smaller solution is \(x = 7 - \sqrt{11}\).
• The larger solution is \(x = 7 + \sqrt{11}\).

Now let's compute the approximate values:

• \(\sqrt{11} \approx 3.3166\)

So,

• The smaller solution: \[ x \approx 7 - 3.3166 \approx 3.6834 \]

• The larger solution: \[ x \approx 7 + 3.3166 \approx 10.3166 \]

Thus, we identify the solutions:

• The smaller of the two solutions is \(x \approx 3.6834\).
• The larger of the two solutions is \(x \approx 10.3166\).

However, if you need the exact values:

• The smaller solution is \(x = 7 - \sqrt{11}\).
• The larger solution is \(x = 7 + \sqrt{11}\).