Answer
The solutions are $x = -\frac{2}{3}$ and $x = 1$.
Work Step by Step
First, we want to rewrite this equation as a quadratic one. The quadratic equation is given as:
$ax^2 + bx + c$, where $a$, $b$, and $c$ are all real numbers.
To do this, we subtract $2$ from each side of the equation:
$3x^2 - x - 2 = 0$
To factor this equation, we want to find which factors, when multiplied, will give us the product of the $a$ and $c$ terms but when added together will give us the $b$ term.
Let's look at possible factors:
$-3$ and $2$
$-6$ and $1$
The first combination will work.
Let's rewrite the equation by splitting the middle term using the factors we found:
$3x^2 - 3x + 2x - 2 = 0$
Group the the first two terms and the second two terms:
$(3x^2 - 3x) + (2x - 2) = 0$
Factor out a $3x$ from the first group and a $2$ from the second group:
$3x(x - 1) + 2(x - 1) = 0$
Group the coefficients together to make one factor whereas the other factor is the common factor:
$(3x + 2)(x - 1) = 0$
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:
$3x + 2 = 0$ or $x - 1 = 0$
Let's look at the first factor:
$3x + 2 = 0$
Subtract $2$ from each side of the equation:
$3x = -2$
Divide each side of the equation by $3$:
$x = -\frac{2}{3}$
Let's look at the second factor:
$x - 1 = 0$
Add $1$ to each side of the equation to solve:
$x = 1$
The solutions are $x = -\frac{2}{3}$ and $x = 1$.
To check our answer, we substitute each of these values into the original equation to see if the two sides equal one another:
$3(-\frac{2}{3})^2 - (-\frac{2}{3}) = 2$
Evaluate exponents first:
$3(\frac{4}{9}) + \frac{2}{3} = 2$
Multiply to simplify:
$\frac{12}{9} + \frac{2}{3} = 2$
Simplify the fraction:
$\frac{4}{3} + \frac{2}{3} = 2$
Add:
$\frac{6}{3} = 2$
Simplify the fraction:
$2 = 2$
The two sides of the equation are equal to one another; therefore, this solution is correct.
Let's check the other solution:
$3(1)^2 - (1) = 2$
Evaluate exponents first:
$3(1) - 1 = 2$
Multiply to simplify:
$3 - 1 = 2$
Subtract:
$2 = 2$
The two sides of the equation are equal to one another; therefore, this solution is correct.
