To determine if \( x = \frac{2}{3} \) is a solution to the quadratic equation \( 9x^2 + 3x - 5 = 6 \), we can substitute \( x = \frac{2}{3} \) into the equation and see if both sides are equal.
First, let's simplify the equation: \[ 9x^2 + 3x - 5 - 6 = 0. \] This simplifies to: \[ 9x^2 + 3x - 11 = 0. \]
Now, substituting \( x = \frac{2}{3} \): \[ 9\left(\frac{2}{3}\right)^2 + 3\left(\frac{2}{3}\right) - 11 \]
Calculating each term:
- \( 9\left(\frac{2}{3}\right)^2 = 9 \cdot \frac{4}{9} = 4 \)
- \( 3\left(\frac{2}{3}\right) = 2 \)
Now, substituting these into the equation: \[ 4 + 2 - 11 = 6 - 11 = -5 \]
Since \( -5 \neq 0 \), the left-hand side does not equal the right side after substitution. Thus, the correct conclusion is:
No, \( x = \frac{2}{3} \) is not a solution because substituting it back into the equation does not yield a true statement.
The closest response that conveys the correct message, albeit with some inaccuracies in the reasoning provided, is: No, \( x = \frac{2}{3} \) is not a solution because substituting it back into the equation results in 6 = 6, which is a false statement. (However, the explanation in this option does not match the actual calculations).
Overall, none of the responses provided are completely correct. The answer is simply that substituting \( \frac{2}{3} \) does not satisfy the equation \( 9x^2 + 3x - 11 \neq 0 \).