Question

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

Answer 1

To determine whether \(-1\) and \(73\) are solutions to the quadratic equation \(3x^2 - 4x - 4 = 3\), we first rewrite the equation in standard form by moving all terms to one side:

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

This simplifies to:

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

Next, we will check if \(-1\) and \(73\) satisfy this equation by substituting each value into the left-hand side of the equation:

1. **Check if \(x = -1\) is a solution:**

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

Since the left-hand side equals \(0\), \(-1\) is indeed a solution to the equation.

2. **Check if \(x = 73\) is a solution:**

\[
3(73)^2 - 4(73) - 7
\]
First, calculate \(73^2\):
\[
73^2 = 5329
\]

Now substitute this value back into the equation:

\[
3(5329) - 4(73) - 7
\]
\[
= 15987 - 292 - 7
\]
\[
= 15987 - 299
\]
\[
= 15988
\]

Since the left-hand side does not equal \(0\), \(73\) is not a solution to the equation.

**Conclusion:**
- \(-1\) is a solution to the equation.
- \(73\) is not a solution to the equation.