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.
-
Begin with the equation: \[ 9x^2 + 3x - 5 = 6 \]
-
Substitute \( x = \frac{2}{3} \): \[ 9\left(\frac{2}{3}\right)^2 + 3\left(\frac{2}{3}\right) - 5 \]
-
Calculate \( 9\left(\frac{2}{3}\right)^2 \): \[ = 9 \cdot \frac{4}{9} = 4 \]
-
Calculate \( 3\left(\frac{2}{3}\right) \): \[ = 2 \]
-
Now substitute these results into the equation: \[ 4 + 2 - 5 \]
-
Simplify: \[ 4 + 2 = 6 \] \[ 6 - 5 = 1 \]
-
Now we need to compare with the right side of the original equation, which is \( 6 \): \[ 1 \neq 6 \]
Since the left-hand side equals \( 1 \) and the right-hand side equals \( 6 \), we can conclude that \( x = \frac{2}{3} \) is not a solution to the equation \( 9x^2 + 3x - 5 = 6 \).
The correct reasoning is: By substituting \( x = \frac{2}{3} \) into the equation, the left-hand side evaluates to \( 1 \), which does not equal \( 6 \). Therefore, \( x = \frac{2}{3} \) is not a solution of the equation.