To determine whether the values \( -1 \) and \( \frac{7}{3} \) are solutions to the quadratic equation \( 3x^2 - 4x - 4 = 3 \), we first rewrite the equation in standard form. We subtract 3 from both sides of the equation:
\[ 3x^2 - 4x - 4 - 3 = 0 \]
This simplifies to:
\[ 3x^2 - 4x - 7 = 0 \]
Now, we will test each value.
Testing \( x = -1 \):
Substituting \( x = -1 \) into the equation:
\[ 3(-1)^2 - 4(-1) - 7 = 0 \]
Calculating the left side:
\[ 3(1) + 4 - 7 = 3 + 4 - 7 = 0 \]
Since the left side equals 0, \( x = -1 \) is indeed a solution.
Testing \( x = \frac{7}{3} \):
Substituting \( x = \frac{7}{3} \) into the equation:
\[ 3\left(\frac{7}{3}\right)^2 - 4\left(\frac{7}{3}\right) - 7 = 0 \]
Calculating \( \left(\frac{7}{3}\right)^2 \):
\[ \left(\frac{7}{3}\right)^2 = \frac{49}{9} \]
Now substitute it into the equation:
\[ 3 \cdot \frac{49}{9} - 4 \cdot \frac{7}{3} - 7 = 0 \]
Calculating \( 3 \cdot \frac{49}{9} \):
\[ \frac{147}{9} \]
Calculating \( -4 \cdot \frac{7}{3} \):
\[ -\frac{28}{3} = -\frac{84}{9} \]
Now we can combine all parts:
\[ \frac{147}{9} - \frac{84}{9} - \frac{63}{9} = 0 \]
Combining the fractions:
\[ \frac{147 - 84 - 63}{9} = \frac{0}{9} = 0 \]
Thus, \( x = \frac{7}{3} \) is also a solution.
In conclusion, both \( -1 \) and \( \frac{7}{3} \) are solutions to the quadratic equation \( 3x^2 - 4x - 4 = 3 \).