Answer
$x = -\frac{4}{3}, -\frac{1}{3}$
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 $9x^2 + 15x + 4 = 0$, $(a)(c)$ is $(9)(4)$, or $36$, but when added together will equal $b$ or $15$. Both factors need to be positive, in this case.
We came up with the following possibilities:
$(a)(c)$ = $(36)(1)$
$b = 37$
$(a)(c)$ = $(12)(3)$
$b = 15$
$(a)(c)$ = $(9)(4)$
$b = 13$
The second pair works. We will use that pair to split the middle term:
$9x^2 + 12x + 3x + 4 = 0$
Now, we can factor by grouping. We group the first two terms together and the last two terms together:
$(9x^2 + 12x) + (3x + 4) = 0$
We see that $3x$ is a common factor for the first group, so let us factor it out:
$3x(3x + 4) + (3x + 4) = 0$
We see that $3x + 4$ is common to both groups, so we put that binomial in parentheses. The other binomial will be $3x + 1$, which is composed of the coefficients sitting in front of the binomials. We now have the two factors:
$(3x + 4)(3x + 1) = 0$
We now set each factor equal to zero and solve:
$3x + 4 = 0$
Subtract $4$ from each side of the equation to isolate constants to the right side of the equation:
$3x = -4$
Divide each side by $3$ to solve for $x$:
$x = -\frac{4}{3}$
Let's set the other factor equal to zero:
$3x + 1 = 0$
Subtract $1$ from each side of the equation to solve for $x$:
$3x = -1$
Divide both sides by $3$ to solve for $x$:
$x = -\frac{1}{3}$
The solution is $x = -\frac{4}{3}, -\frac{1}{3}$.
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 $x = -\frac{4}{3}$ first:
$9(-\frac{4}{3})^2 + 15(-\frac{4}{3}) + 4 = 0$
Evaluate the exponent first:
$9(\frac{16}{9}) + 15(-\frac{4}{3}) + 4 = 0$
Multiply next, according to order of operations:
$\frac{(16)(9)}{9} - \frac{(4)(15)}{3} + 4 = 0$
Cross-cancel factors to simplify:
$16 - 20 + 4 = 0$
Subtract or add from left to right, according to order of operations:
$-4 + 4 = 0$
Add to simplify:
$0 = 0$
The left and right sides are equal; therefore, this solution is correct.
Let's check $x = -\frac{1}{3}$:
$9(-\frac{1}{3})^2 + 15(-\frac{1}{3}) + 4 = 0$
Evaluate the exponent first:
$9(\frac{1}{9}) + 15(-\frac{1}{3}) + 4 = 0$
Multiply next, according to order of operations:
$9(\frac{1}{9}) - \frac{(1)(15)}{3} + 4 = 0$
Cross-cancel factors to simplify:
$1 - 5 + 4 = 0$
Subtract or add from left to right, according to order of operations:
$-4 + 4 = 0$
Add to simplify:
$0 = 0$
The left and right sides are equal; therefore, this solution is correct.
