Use the table to answer the question. x Substituted Evaluate True Statement? −23 0 12 For the quadratic equation x2+20x=69 , complete the table by testing each of the given values to determine whether it is a solution. Identify which one of these values is in the solution set. (1 point) Only x=blank is in the solution set.

Answer 1

To determine which value is in the solution set of the quadratic equation \(x^2 + 20x = 69\), we first rearrange the equation to standard form:

\[ x^2 + 20x - 69 = 0 \]

Next, we will evaluate this equation for the given values: \(x = -23\), \(x = 0\), and \(x = 12\).

1. For \(x = -23\): \[ (-23)^2 + 20(-23) - 69 = 529 - 460 - 69 = 0 \] This is a true statement.

2. For \(x = 0\): \[ (0)^2 + 20(0) - 69 = 0 - 69 = -69 \] This is not a true statement.

3. For \(x = 12\): \[ (12)^2 + 20(12) - 69 = 144 + 240 - 69 = 315 \] This is not a true statement.

After evaluating all three values, we find only \(x = -23\) results in the equation being equal to zero.

Thus, the answer is:

Only \(x = -23\) is in the solution set.

Answer 2

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=blank. The larger of the two solutions is x=blank.

Answer 3

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

1. Rearranging the equation gives: \[ x - 7 = \pm \sqrt{11} \]

2. This results in two equations: \[ x - 7 = \sqrt{11} \quad \text{and} \quad x - 7 = -\sqrt{11} \]

3. Solving each equation for \(x\):

   • First equation: \[ x = 7 + \sqrt{11} \]

   • Second equation: \[ x = 7 - \sqrt{11} \]

Now we have the two potential solutions:

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

Next, we can evaluate these solutions using approximate decimal values:

• The approximate value of \(\sqrt{11} \approx 3.3166\).

Therefore:

• Larger solution: \[ x = 7 + \sqrt{11} \approx 7 + 3.3166 \approx 10.3166 \]

• Smaller solution: \[ x = 7 - \sqrt{11} \approx 7 - 3.3166 \approx 3.6834 \]

Finally, we can fill in the blanks:

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

Thus, the final answers are:

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

Answer 4

How many solutions would the equation x2=−9 have? You do not need to solve for x.(1 point) The equation would have blank solution(s).

Answer 5

The equation \(x^2 = -9\) cannot have real solutions because the square of a real number cannot be negative. However, if we consider complex numbers, we can find solutions.

For the equation \(x^2 = -9\), we can express it as:

\[ x = \pm \sqrt{-9} = \pm 3i \]

Thus, in the context of complex numbers, there are 2 solutions: \(3i\) and \(-3i\).

Therefore, the statement would be:

The equation would have 2 solutions.