Answer
The solutions are $h = 0$, $h = 7$, and $h = -2$.
Work Step by Step
First, we collect all terms on the left side of the equation:
$12h^3 - 60h^2 - 168h = 0$
Next, we factor out what is common in both terms:
$12h(h^2 - 5h - 14) = 0$
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:
$-7$ and $2$
$-14$ and $1$
The first combination will work.
Let's write the equation in factor form:
$12h(h - 7)(h + 2) = 0$
According to the zero product property, if the product of factors equals $0$, then each of the factors can be $0$; therefore, we can set each of these factors equal to $0$ and solve.
Let's set the first factor equal to $0$:
$12h = 0$
Divide each side of the equation by $12$:
$h = 0$
For the second factor, we add $7$ to each side of the equation:
$h = 7$
For the third factor, we subtract $2$ from each side of the equation:
$h = -2$
The solutions are $h = 0$, $h = 7$, and $h = -2$.
To check our answers, we substitute each of these values into the original equation to see if the two sides equal one another.
Let's look at the first factor:
$12(0)^3 - 60(0)^2 = 168(0)$
Evaluate exponents first:
$12(0) - 60(0) = 168(0)$
Multiply to simplify:
$0 - 0 = 0$
Subtract:
$0 = 0$
The two sides of the equation are equal to one another; therefore, this solution is correct.
Let's check the second solution:
$12(7)^3 - 60(7)^2 = 168(7)$
Evaluate exponents first:
$12(343) - 60(49) = 168(7)$
Multiply to simplify:
$4116 - 2940 = 1176$
Subtract:
$1176 = 1176$
The two sides of the equation are equal to one another; therefore, this solution is correct.
Let's check the third solution:
$12(-2)^3 - 60(-2)^2 = 168(-2)$
Evaluate exponents first:
$12(-8) - 60(4) = 168(-2)$
Multiply to simplify:
$-96 - 240 = -336$
Subtract:
$-336 = -336$
The two sides of the equation are equal to one another; therefore, this solution is correct.