Answer
$x = -\frac{2}{3}, -2$
Work Step by Step
To factor a quadratic equation in the form $ax^2 + bx + c = 0$, we look at factors of the product of $a$ and $c$ such that, when added together, equal $b$.
For the equation $3x^2 + 8x + 4 = 0$, $(a)(c)$ is $(3)(4)$, or $12$, but when added together will equal $b$ or $8$. Both factors need to be positive, in this case.
We came up with the following possibilities:
$(a)(c)$ = $(12)(1)$
$b = 13$
$(a)(c)$ = $(6)(2)$
$b = 8$
$(a)(c)$ = $(4)(3)$
$b = 7$
The second pair works. We will use that pair to split the middle term:
$3x^2 + 6x + 2x + 4 = 0$
Now, we can factor by grouping. We group the first two terms together and the last two terms together:
$(3x^2 + 6x) + (2x + 4) = 0$
We see that $3x$ is a common factor for the first group and $2$ is a common factor for the second group, so let us factor these out:
$3x(x + 2) + 2(x + 2) = 0$
We see that $x + 2$ is common to both groups, so we put that binomial in parentheses. The other binomial will be $3x + 2$, which is composed of the coefficients sitting in front of the binomials. We now have the two factors:
$(3x + 2)(x + 2) = 0$
We now set each factor equal to zero and solve:
$3x + 2 = 0$
Subtract $2$ from each side of the equation to isolate constants to the right side of the equation:
$3x = -2$
Divide each side by $3$ to solve for $x$:
$x = -\frac{2}{3}$
Let's set the other factor equal to zero:
$x + 2 = 0$
Subtract $2$ from each side of the equation to solve for $x$:
$x = -2$
The solution is $x = -\frac{2}{3}, -2$.
To check if our solutions are correct, we plug our solutions back into the equation to see if the left and right sides equal one another.
Let's plug in $x = -\frac{2}{3}$ first:
$3(-\frac{2}{3})^2 + 8(-\frac{2}{3}) + 4 = 0$
Evaluate the exponent first:
$3(\frac{4}{9})^2 + 8(-\frac{2}{3}) + 4 = 0$
Multiply next, according to order of operations:
$\frac{12}{9} - \frac{16}{3} + 4 = 0$
Convert the fractions so that they have the same denominator. In this case, we can convert the second fraction so that its denominator is $9$:
$\frac{12}{9} - \frac{48}{9} + 4 = 0$
Subtract the fractions, according to order of operations:
$-\frac{36}{9} + 4 = 0$
Simplify the fraction:
$-4 + 4 = 0$
Add to simplify:
$0 = 0$
The left and right sides are equal; therefore, this solution is correct.
Let's check $x = -2$:
$3(-2)^2 + 8(-2) + 4 = 0$
Evaluate exponents first, according to order of operations:
$3(4) + 8(-2) + 4 = 0$
Multiply next, according to order of operations:
$12 - 16 + 4 = 0$
Add or subtract from left to right:
$-4 + 4 = 0$
Add to simplify:
$0 = 0$
The left and right sides are equal; therefore, this solution is correct.
