Answer
$x = -5, -4$
Work Step by Step
To factor a quadratic polynomial 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 $x^2 + 9x + 20 = 0$, $(a)(c)$ is $(1)(20)$, or $20$, but when added together will equal $b$ or $9$. Both factors need to be positive in this case.
We came up with the following possibilities:
$(a)(c)$ = $(20)(1)$
$b = 21$
$(a)(c)$ = $(10)(2)$
$b = 12$
$(a)(c)$ = $(5)(4)$
$b = 9$
The third pair, $5$ and $4$, will work. Let us factor the polynomial incorporating these factors:
$(x + 5)(x + 4) = 0$
According to the zero product property, if the product of two factors $a$ and $b$ equals zero, then either $a$ is zero, $b$ is zero, or both equal zero. Therefore, we can set each factor equal to zero:
The first factor:
$x + 5 = 0$
Subtract $5$ from each side to solve for $x$:
$x = -5$
The second factor:
$x + 4 = 0$
Subtract $4$ from each side to solve for $x$:
$x = -4$
The solution is $x = -5, -4$.
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 $x = -5$ first:
$(-5)^2 + 9(-5) + 20 = 0$
Multiply first, according to order of operations:
$25 - 45 + 20 = 0$
Now add and subtract from left to right:
$-20 + 20 = 0$
Add to simplify:
$0 = 0$
The left and right sides are equal; therefore, this solution is correct.
Let's check $x = -4$:
$(-4)^2 + 9(-4) + 20 = 0$
Multiply first, according to order of operations:
$16 - 36 + 20 = 0$
Now add and subtract from left to right:
$-20 + 20 = 0$
Add to simplify:
$0 = 0$
The left and right sides are equal; therefore, this solution is also correct.
