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Determine whether the values −1

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

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.

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

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

1 answer

To determine whether the values \( x = -1 \) and \( x = 73 \) are solutions to the quadratic equation \( 3x^2 - 4x - 4 = 3 \), we first need to rewrite the equation in standard form:

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

This simplifies to:

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

Now, we will check each value separately.

Checking \( x = -1 \):

Substituting \( x = -1 \) into the equation:

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

Checking \( x = 73 \):

Substituting \( x = 73 \) into the equation:

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

Calculating \( 73^2 \):

\[ 73^2 = 5329 \]

Now substituting back:

\[ 3(5329) - 4(73) - 7 = 15987 - 292 - 7 \] \[ = 15987 - 299 = 15688 \]

This is not equal to 0.

Conclusion:

  • \( x = -1 \) is a solution since it makes the equation true.
  • \( x = 73 \) is not a solution.

Thus, the correct response is:

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