Answer
$t = 10$ or $t = 2$
Work Step by Step
We have a quadratic equation in the form of $ax^2 + bx + c$, where $a$, $b$, and $c$ are all real numbers.
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:
$-5$ and $-4$
$-10$ and $-2$
It looks like the second combination will work. Let's put the factors together:
$(t - 10)(t - 2) = 0$
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:
$t - 10 = 0$ or $t - 2 = 0$
Add to solve:
$t = 10$ or $t = 2$
To check our answer, we substitute each of these values into the original equation to see if the two sides equal one another:
$(10)^2 - 12(10) + 20 = 0$
Multiply to simplify:
$100 - 120 + 20 = 0$
Subtract or add from left to right:
$-20 + 20 = 0$
Add once again:
$0 = 0$
The two sides of the equation are equal to one another; therefore, this solution is correct. Let's check the other solution:
$(2)^2 - 12(2) + 20 = 0$
Multiply to simplify:
$4 - 24 + 20 = 0$
Subtract or add from left to right:
$-20 + 20 = 0$
Add once again:
$0 = 0$
The two sides of the equation are equal to one another; therefore, this solution is correct.
