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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=.

1 answer

To solve the quadratic equation \((x - 7)^2 = 11\), we will first find the solutions:

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

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

Now we need to identify and complete the table with the provided values and check if they satisfy the equation.

Let’s evaluate:

  1. Value \(7 - \sqrt{11}\): \[ (7 - \sqrt{11} - 7)^2 = (-\sqrt{11})^2 = 11 \quad \text{(valid solution)} \]

  2. Value \(7\): \[ (7 - 7)^2 = 0 \neq 11 \quad \text{(not a solution)} \]

  3. Value \(11 - \sqrt{7}\): \[ (11 - \sqrt{7} - 7)^2 = (4 - \sqrt{7})^2 \neq 11 \quad \text{(not a solution)} \]

  4. Value \(7 + \sqrt{11}\): \[ (7 + \sqrt{11} - 7)^2 = (\sqrt{11})^2 = 11 \quad \text{(valid solution)} \]

  5. Value \(11 + \sqrt{7}\): \[ (11 + \sqrt{7} - 7)^2 = (4 + \sqrt{7})^2 \neq 11 \quad \text{(not a solution)} \]

Now, we have identified the valid solutions to the equation \( (x - 7)^2 = 11\):

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

To summarize:

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