Answer
The solutions are $x = 10$ and $x = -6$.
Work Step by Step
First, we factor out what is common in both terms:
$2(x^2 - 4x - 60) = 0$
Simplify by dividing both sides of the equation by $2$:
$x^2 - 4x - 60 = 0$
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:
$-10$ and $6$
$-12$ and $5$
$-60$ and $1$
The first combination will work.
Let's write the equation in factor form:
$(x - 10)(x + 6) = 0$
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.
Let's set the first factor equal to $0$:
$x - 10 = 0$
Add $10$ to each side of the equation:
$x = 10$
For the second factor, we subtract $6$ from each side of the equation:
$x = -6$
The solutions are $x = 10$ and $x = -6$.
To check our answers, we substitute each of these values into the original equation to see if the two sides equal one another:
$2(10)^2 - 8(10) - 120 = 0$
Evaluate exponents first:
$2(100) - 8(10) - 120 = 0$
Multiply to simplify:
$200 - 80 - 120 = 0$
Subtract from left to right:
$0 = 0$
The two sides of the equation are equal to one another; therefore, this solution is correct. Let's check the other solution:
$2(-6)^2 - 8(-6) - 120 = 0$
Evaluate exponents first:
$2(36) - 8(-6) - 120 = 0$
Multiply to simplify:
$72 + 48 - 120 = 0$
Subtract from left to right:
$0 = 0$
The two sides of the equation are equal to one another; therefore, this solution is correct.
