Answer
The solutions are $x = 0$ and $x = \frac{4}{3}$.
Work Step by Step
First, we factor out what is common in both terms:
$2x(3x - 4) = 0$
According to the zero product property, if the product of two factors equals $0$, then either factor can be $0$; therefore, we can set each of these factors equal to $0$ and solve.
Let's set the first factor equal to $0$:
$2x = 0$
Divide each side of the equation by $2$:
$x = 0$
For the second factor, we add $4$ to each side of the equation:
$3x = 4$
Divide each side by $3$ to solve:
$x = \frac{4}{3}$
The solutions are $x = 0$ and $x = \frac{4}{3}$.
To check our answers, we substitute each of these values into the original equation to see if the two sides equal one another:
$6(0)^2 - 8(0) = 0$
Multiply to simplify:
$0 - 0 = 0$
Subtract:
$0 = 0$
The two sides of the equation are equal to one another; therefore, this solution is correct. Let's check the other solution:
$6(\frac{4}{3})^2 - 8(\frac{4}{3}) = 0$
Evaluate the exponent first:
$6(\frac{16}{9}) - 8(\frac{4}{3}) = 0$
Multiply to simplify:
$\frac{96}{9} - \frac{32}{3} = 0$
Simplify the fractions:
$\frac{32}{3} - \frac{32}{3} = 0$
Subtract:
$0 = 0$
The two sides of the equation are equal to one another; therefore, this solution is correct.
