Question

Ben is trying to determine if x = 2/3 is a solution to the quadratic equation 9x^2 + 3x − 5 = 6. Which explanation below demonstrates the correct reasoning?
A) No, x = 2/3 is not a solution because substituting it back into the equation results in 1 = 6, which is a false statement.
B) Yes, x = 2/3 is a solution because substituting it back into the equation results in 6 = 6, which is a true statement.
C) No, x = 2/3 is not a solution because substituting it back into the equation results in 6 = 6, which is a false statement.
D) Yes, x = 2/3 is a solution because substituting it back into the equation results in 1 = 6, which is a true statement.

Answer 1

To determine if \( x = \frac{2}{3} \) is a solution to the equation \( 9x^2 + 3x - 5 = 6 \), we can substitute \( \frac{2}{3} \) into the left side of the equation and see if it equals 6.

1. Substitute \( x = \frac{2}{3} \) into the left side of the equation:
\[
9\left(\frac{2}{3}\right)^2 + 3\left(\frac{2}{3}\right) - 5
\]

2. Calculate \( 9\left(\frac{2}{3}\right)^2 \):
\[
\left(\frac{2}{3}\right)^2 = \frac{4}{9} \implies 9 \cdot \frac{4}{9} = 4
\]

3. Calculate \( 3\left(\frac{2}{3}\right) \):
\[
3 \cdot \frac{2}{3} = 2
\]

4. Combine these results:
\[
4 + 2 - 5 = 6 - 5 = 1
\]

5. The left side evaluates to 1, while the right side of the original equation is 6.

Since \( 1 \neq 6 \), we conclude that \( x = \frac{2}{3} \) is not a solution.

Thus, the correct explanation is:

**A)** No, \( x = \frac{2}{3} \) is not a solution because substituting it back into the equation results in \( 1 = 6 \), which is a false statement.