Determine whether the values 3 and 4 are solutions to the quadratic equation x^2−7x+12=0

Question

, 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)

Answer 1

To determine whether 3 and 4 are solutions to the quadratic equation \( x^2 - 7x + 12 = 0 \), we can substitute each value into the equation and check if each results in equality.

1. Substituting \( x = 3 \): \[ (3)^2 - 7(3) + 12 = 9 - 21 + 12 = 0 \] Since the left side equals 0, \( x = 3 \) is a solution.

2. Substituting \( x = 4 \): \[ (4)^2 - 7(4) + 12 = 16 - 28 + 12 = 0 \] Since the left side equals 0, \( x = 4 \) is also a solution.

Thus, both \( x = 3 \) and \( x = 4 \) are solutions to the equation.

The correct answer is:

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

Answer 2

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.

Answer 3

To determine whether the values -23, 0, and 12 are solutions to the quadratic equation \(x^2 + 20x = 69\), we first rewrite the equation in standard form:

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

Now we will substitute each value into the left side of the equation and see if it equals 0.

1. Substituting \(x = -23\): \[ (-23)^2 + 20(-23) - 69 = 529 - 460 - 69 = 0 \] This simplifies to: \[ 529 - 460 - 69 = 529 - 529 = 0 \] So, \(x = -23\) is a solution.

2. Substituting \(x = 0\): \[ (0)^2 + 20(0) - 69 = 0 + 0 - 69 = -69 \] This does not equal 0, so \(x = 0\) is not a solution.

3. Substituting \(x = 12\): \[ (12)^2 + 20(12) - 69 = 144 + 240 - 69 = 144 + 240 - 69 = 315 \] This does not equal 0, so \(x = 12\) is not a solution.

Now we compile the results:

• For \(x = -23\): True (it equals 0)
• For \(x = 0\): False (it does not equal 0)
• For \(x = 12\): False (it does not equal 0)

Thus, the only value that is in the solution set is:

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