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Use the table to answer the question.

x Substituted Evaluate True Statement?
7−11−−√

7
11−7–√

7+11−−√

11+7–√

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.

(1 point)
The smaller of the two solutions is x=
. The larger of the two solutions is x=
.

3 answers

To solve the quadratic equation \((x - 7)^2 = 11\), we first need to find the values of \(x\) that satisfy this equation.

  1. Start by taking the square root of both sides: \[ x - 7 = \pm \sqrt{11} \]

  2. Now, solve for \(x\): \[ x = 7 + \sqrt{11} \quad \text{and} \quad x = 7 - \sqrt{11} \]

These two values are the solutions to the equation.

Now, approximate \(\sqrt{11}\): \[ \sqrt{11} \approx 3.3166 \]

Thus, we can find the two solutions:

  • For \(x = 7 + \sqrt{11}\): \[ x \approx 7 + 3.3166 \approx 10.3166 \]
  • For \(x = 7 - \sqrt{11}\): \[ x \approx 7 - 3.3166 \approx 3.6834 \]

Based on the approximations, we conclude:

  • The smaller solution is \(x \approx 3.6834\).
  • The larger solution is \(x \approx 10.3166\).

Thus, filling in the blanks from your question: The smaller of the two solutions is \(x = 7 - \sqrt{11}\).
The larger of the two solutions is \(x = 7 + \sqrt{11}\).

what does '\sqrt' mean?

The symbol "\(\sqrt{}\)" represents the square root operation in mathematics.

When you see \(\sqrt{x}\), it means "the square root of \(x\)." In other words, it is the value that, when multiplied by itself, gives \(x\). For example:

  • \(\sqrt{4} = 2\) because \(2 \times 2 = 4\).
  • \(\sqrt{9} = 3\) because \(3 \times 3 = 9\).
  • If \(x\) is a negative number, like \(\sqrt{-1}\), the result is not a real number (in the context of real numbers) but is considered an imaginary number, represented as \(i\).

Please let me know if you need further clarification or examples!