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

Answer 1

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

1. For \( x = 3 \): \[ 3^2 - 7 \cdot 3 + 12 = 9 - 21 + 12 = 0 \] Thus, \( x = 3 \) is a solution.

2. For \( x = 4 \): \[ 4^2 - 7 \cdot 4 + 12 = 16 - 28 + 12 = 0 \] Thus, \( x = 4 \) is also a solution.

Since both values satisfy the equation, the correct answer is: Option #3: Both x=3 and x=4 are solutions.