Question

Solution Sets of Quadratic Equations Quick Check
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Question
Determine whether the values −1 and 73 are solutions to the quadratic equation 3x2−4x−4=3.(1 point)
Responses

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

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=−1 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 need to simplify the equation.

We can rewrite the equation as follows:
\[
3x^2 - 4x - 4 - 3 = 0
\]
This simplifies to:
\[
3x^2 - 4x - 7 = 0
\]

Now we will substitute \( x = -1 \) and \( x = 73 \) into the simplified equation.

**1. Testing \( x = -1 \):**
\[
3(-1)^2 - 4(-1) - 7 = 3(1) + 4 - 7 = 3 + 4 - 7 = 0
\]
So, \( x = -1 \) is a solution.

**2. Testing \( x = 73 \):**
\[
3(73)^2 - 4(73) - 7
\]
Calculating \( 3(73)^2 \):
\[
3(5329) = 15987
\]
Calculating \( -4(73) \):
\[
-4(73) = -292
\]
Now combining these:
\[
15987 - 292 - 7 = 15688 \neq 0
\]
Thus, \( x = 73 \) is not a solution.

In conclusion, the correct response is:
**Only \( x = -1 \) is a solution.**