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.

Question

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

Answer 1

To determine if 3 and 4 are solutions of the quadratic equation \( x^2 - 7x + 12 = 0 \), we can substitute these values into the equation and check if the left-hand side equals 0.

1. Testing \( x = 3 \):

\[ x^2 - 7x + 12 = 3^2 - 7(3) + 12 \] \[ = 9 - 21 + 12 \] \[ = 9 - 21 + 12 = 0 \]

Since the equation equals 0, \( x = 3 \) is a solution.

2. Testing \( x = 4 \):

\[ x^2 - 7x + 12 = 4^2 - 7(4) + 12 \] \[ = 16 - 28 + 12 \] \[ = 16 - 28 + 12 = 0 \]

Since the equation also equals 0, \( x = 4 \) is a solution.

Both values satisfy the equation.

Therefore, both \( x = 3 \) and \( x = 4 \) are solutions.

The correct answer is:

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