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Determine whether the values 3 and 4 are solutions to the quadratic equation x2−7x+12=0 , and then select the correct answer from the following options.

Option #1: Only x=3 is a solution.

Option #2: Only x=4 is a solution.

Option #3: Both x=3 and x=4 are solutions.

Option #4: Neither x=3 nor x=4 is a solution.

(1 point)
Option #

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=
is in the solution set.

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=
.
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).

Assuming an equation with one side as a squared variable expression and the other side as a numeric expression, which of the following statements is correct?

Statement #1: If the numeric expression is greater than zero, there are two solutions.

Statement #2: If the numeric expression is greater than zero, there is one solution.

Statement #3: If the numeric expression is greater than zero, there are no solutions.

(1 point)
Statement #
is correct.

1 answer

Let's tackle each part of your query step by step.

  1. Determine if 3 and 4 are solutions to the equation \( x^2 - 7x + 12 = 0 \): To check if 3 is a solution: \[ 3^2 - 7(3) + 12 = 9 - 21 + 12 = 0 \] This is true, so \( x = 3 \) is a solution.

    Now checking if 4 is a solution: \[ 4^2 - 7(4) + 12 = 16 - 28 + 12 = 0 \] This is also true, so \( x = 4 \) is a solution.

    Since both 3 and 4 are solutions, the correct option is: Option #3: Both \( x=3 \) and \( x=4 \) are solutions.

  1. For the quadratic equation \( x^2 + 20x = 69 \), we will evaluate the table: Rearranging gives us \( x^2 + 20x - 69 = 0 \). We will substitute the values and check.

    Substituting -23: \[ (-23)^2 + 20(-23) = 529 - 460 = 69 \quad \text{(True)} \]

    Substituting 0: \[ 0^2 + 20(0) = 0 \quad \text{(False)} \]

    Substituting 12: \[ 12^2 + 20(12) = 144 + 240 = 384 \quad \text{(False)} \]

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

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

  1. For the quadratic equation \( (x-7)^2 = 11 \): Rearranging gives \( (x - 7)^2 - 11 = 0 \). The square root yields two solutions: \[ x - 7 = \pm \sqrt{11} \] Thus, we have: \[ x = 7 + \sqrt{11} \quad \text{and} \quad x = 7 - \sqrt{11} \]

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

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

  1. How many solutions does the equation \( x^2 = -9 \) have? A quadratic equation has no real solutions if the expression on the other side of the equation is negative. Since -9 is negative, there are no real solutions.

    The answer is: The equation would have 0 solution(s).

  1. Evaluating the statements regarding an equation with one side as a squared variable expression:

    • Statement #1: If the numeric expression is greater than zero, there may be two solutions (for example \( x^2 = 1 \)).
    • Statement #2: This is incorrect; it can have two solutions, not one.
    • Statement #3: This is incorrect; if the expression is greater than zero, it can have solutions depending on the case.

    The correct statement is: Statement #1 is correct.