Question

True or False: The values -1 and 7/3
are both solutions to the quadratic equation 3x2−4x−4=3
.

Answer 1

To determine if 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 by subtracting 3 from both sides:

\[
3x^2 - 4x - 4 - 3 = 0
\]
which simplifies to:
\[
3x^2 - 4x - 7 = 0
\]

Now we will check each value:

1. For \( x = -1 \):
\[
3(-1)^2 - 4(-1) - 7 = 3(1) + 4 - 7 = 3 + 4 - 7 = 0
\]
Since the left side equals the right side (0 = 0), \( x = -1 \) is a solution.

2. For \( x = \frac{7}{3} \):
\[
3\left(\frac{7}{3}\right)^2 - 4\left(\frac{7}{3}\right) - 7
\]
Calculating \( 3\left(\frac{7}{3}\right)^2 \):
\[
3\left(\frac{49}{9}\right) = \frac{147}{9}
\]

Now calculate \( -4\left(\frac{7}{3}\right) \):
\[
-4 \cdot \frac{7}{3} = -\frac{28}{3}
\]

To combine the terms, we convert \( -\frac{28}{3} \) to have a common denominator:
\[
-\frac{28}{3} = -\frac{84}{9}
\]

Now substitute back into the equation:
\[
\frac{147}{9} - \frac{84}{9} - 7 = \frac{147 - 84}{9} - 7 = \frac{63}{9} - 7
\]
\[
= \frac{63}{9} - \frac{63}{9} = 0
\]
Since the left side equals the right side (0 = 0), \( x = \frac{7}{3} \) is also a solution.

Therefore, the final answer is **True**: both -1 and \( \frac{7}{3} \) are solutions to the given quadratic equation.