Solution Sets of Quadratic Equations Quick Check

Question

1 of 51 of 5 Items
Question
Determine whether the values −1 and 73 are solutions to the quadratic equation 3x2−4x−4=3.(1 point)
Responses

Both x=−1 and x=73 are solutions.
Both x equals negative 1 and x equals Start Fraction 7 over 3 End Fraction are solutions.

Only x=73 is a solution.
Only x equals Start Fraction 7 over 3 End Fraction is a solution.

Neither x=−1 nor x=73 is a solution.Neither x equals negative 1 nor x equals Start Fraction 7 over 3 End Fraction is a solution.

Only x=−1 is a solution.

Answer 1

To determine whether the values \( -1 \) and \( 73 \) are solutions to the quadratic equation \( 3x^2 - 4x - 4 = 3 \), we first need to simplify the equation.

We can rewrite the equation as follows: \[ 3x^2 - 4x - 4 - 3 = 0 \] This simplifies to: \[ 3x^2 - 4x - 7 = 0 \]

Now we will substitute \( x = -1 \) and \( x = 73 \) into the simplified equation.

1. Testing \( x = -1 \): \[ 3(-1)^2 - 4(-1) - 7 = 3(1) + 4 - 7 = 3 + 4 - 7 = 0 \] So, \( x = -1 \) is a solution.

2. Testing \( x = 73 \): \[ 3(73)^2 - 4(73) - 7 \] Calculating \( 3(73)^2 \): \[ 3(5329) = 15987 \] Calculating \( -4(73) \): \[ -4(73) = -292 \] Now combining these: \[ 15987 - 292 - 7 = 15688 \neq 0 \] Thus, \( x = 73 \) is not a solution.

In conclusion, the correct response is: Only \( x = -1 \) is a solution.