Determine whether the values −1 and 73 are solutions to the quadratic equation 3x2−4x−4=3.(1 point)

Question

Responses

Only x=−1 is a solution.
Only x equals negative 1 is a solution.

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.

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=73 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 simplify the equation by moving \(3\) to the left side:

\[ 3x^2 - 4x - 4 - 3 = 0 \]

This simplifies to:

\[ 3x^2 - 4x - 7 = 0 \]

Now we will test both \(x = -1\) and \(x = 73\) as possible solutions by substituting them into the equation.

Testing \(x = -1\):

\[ 3(-1)^2 - 4(-1) - 7 = 3(1) + 4 - 7 = 3 + 4 - 7 = 0 \]

So, \(x = -1\) is a solution.

Testing \(x = 73\):

\[ 3(73)^2 - 4(73) - 7 \]

Calculating \(3(73)^2\):

\[ 3(5329) = 15987 \]

Calculating \(-4(73)\):

\[ -4(73) = -292 \]

Now substituting back in:

\[ 15987 - 292 - 7 = 15987 - 299 = 15688 \neq 0 \]

So, \(x = 73\) is not a solution.

Based on this evaluation, we conclude that:

Only \(x = -1\) is a solution.