Answer
The solution is $x = \frac{4}{3}$.
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 add $11$ to each side of the equation:
$9x^2 - 24x + 16 = 0$
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:
$-12$ and $-12$
$-144$ and $-1$
The first combination will work.
Let's rewrite the equation by splitting the middle term using the factors we found:
$9x^2 - 12x - 12x + 16 = 0$
Group the the first two terms and the second two terms:
$(9x^2 - 12x) + (-12x - 16) = 0$
Factor out a $3x$ from the first group and a $-4$ from the second group:
$3x(3x - 4) - 4(3x - 4) = 0$
Group the factors together:
$(3x - 4)(3x - 4) = 0$
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. In this case, we only have to use one factor because the two factors are the same:
$3x - 4 = 0$
Add $4$ to each side of the equation:
$3x = 4$
Divide each side of the equation by $3$:
$x = \frac{4}{3}$
The solution is $x = \frac{4}{3}$.
To check our answer, we substitute this value into the original equation to see if the two sides equal one another:
$9(\frac{4}{3})^2 - 24(\frac{4}{3}) + 5 = -11$
Evaluate exponents first:
$9(\frac{16}{9}) - 24(\frac{4}{3}) + 5 = -11$
Multiply to simplify:
$\frac{144}{9} - \frac{96}{3} - 5 = -11$
Simplify the fraction:
$\frac{48}{3} - \frac{96}{3} + 5 = -11$
Subtract first:
$-\frac{48}{3} + 5 = -11$
Simplify the fraction:
$-16 + 5 = -11$
Add:
$-11 = -11$
The two sides of the equation are equal to one another; therefore, this solution is correct.
