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Q3

x Substituted Evaluate True Statement
7-square root of 11
7
11-square root of 7
7+square root of 11
11+square root of 7
Question
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 start by taking the square root of both sides:

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

This leads to two equations:

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

Now we solve for \(x\) in both cases:

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

Now, we can evaluate these two solutions:

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

Now, let's approximate \(\sqrt{11}\) for better understanding. The approximation of \(\sqrt{11}\) is about 3.3166.

So, we can find the values:

  1. \(x = 7 - 3.3166 \approx 3.6834\)
  2. \(x = 7 + 3.3166 \approx 10.3166\)

Thus, we can fill in the blanks:

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

Now back to your question:

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

How many solutions would the equation x^2=-9 have? You do not need to solve for x.(1 point)
The equation would have____solution(s).

The equation \(x^2 = -9\) involves squaring \(x\), which results in a non-negative value. Since the right side of the equation is negative (-9), there are no real solutions because the square of any real number cannot equal a negative number.

However, if we consider complex numbers, \(x^2 = -9\) would have two solutions, given by \(x = 3i\) and \(x = -3i\), where \(i\) is the imaginary unit.

To directly answer your question:

The equation would have no real solutions but two complex solutions. If you're looking for real solutions specifically, the answer would be 0 solutions.