Answer
The solutions are $x = 7, -2$.
Work Step by Step
To factor a quadratic trinomial in the form $x^2 + bx + c = 0$, we look at factors of $c$ that when added together, equal $b$.
For the equation $x^2 - 5x - 14 = 0$, we look for the factors of $-14$ that when added together will equal $b$ or $-5$:
$-14=(-14)(1)$
$-14+1 = -13$
$-14=(14)(-1)$
$-14+(-1) = 13$
$-14=(-7)(2)$
$-7+2 = -5$
$-14=(7)(-2)$
$7+(-2) = -5$
The third pair gives a sum of $-5$, which is equal to $b$. Thus, the factored form of the trinomial is $(x-7)(x+2)$. Hence, the equation above can be rewritten as:
$$(x - 7)(x + 2) = 0$$
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 then solve each equation.
First factor:
$x - 7 = 0$
$x = 7$
Second factor:
$x + 2 = 0$
$x = -2$
Thus, the solutions are $x = 7, -2$.
