Answer
$x = 7, -2$
Work Step by Step
Let's rewrite the equation so that all terms are on the left side of the equation:
$x^2 - 5x - 14 = 0$
To factor a quadratic trinomial in the form $x^2 + bx + c$, we look at factors $c$ such that, when added together equals $b$.
For the trinomial $x^2 - 5x - 14$, $c=-14$ so we have to look for factors of $-14$ that when added together will equal $b$ or $-5$. One of the factors needs to be positive and the other one negative, but the positive factor should be the one with the greater absolute value. This is because a negative number multiplied with a positive number will equal a negative number; however, when a negative number is added to a positive number, the result can be either negative or positive, depending on which number has the greater absolute value.
The possibilities are:
$-14=(-14)(1)$
$-14+1 = -13$
$-14=(-7)(2)$
$-7+2 = -5$
The second pair, $-7$ and $2$, is the one we are looking for.
Thus, the factored form of the trinomial is $(x - 7)(x + 2)$ and the equation above is equivalent to:
$$(x-7)(x+2)=0$$
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 and solve each one.
For the first factor:
$x - 7 = 0$
$x = 7$
For the second factor:
$x + 2 = 0$
$x = -2$
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.
When $x = 7$:
$x^2=5x+14\\
7^2=5(7)+14\\
49=35+14\\
49=49$
For $x = -2$:
$x^2=5x+14\\
(-2)^2 = 5(-2)+ 14\\
4=-10+14\\
4=4$
Thus, $7$ and $-2$ are the solutions to the given equation.
