Is there a greatest common factor between all of the terms in the given expression? If so, you should factor that out first.
G
Practice makes perfect
We want to find the sum of the roots of the given equation. In order to do that, we will factor the equation and then use the Zero Product Property to solve it.
Factoring
Let's start by rewriting the equation in such way that all the terms are on the left side of the equality sign.
x^2+3x=54
⇕
x^2+3x-54=0
Now, to factor a trinomial with a leading coefficient of 1, think of the process as multiplying two binomials in reverse. Let's take a look at the constant term.
x^2+3x -54=0
In this case, we have -54. This is a negative number, so for the product of the constant terms in the factors to be negative, these constants must have the opposite sign (one positive and one negative).
Factor Constants
Product of Constants
1 and - 54
- 54
-1 and 54
- 54
2 and - 27
- 54
-2 and 27
- 54
3 and - 18
- 54
-3 and 18
- 54
6 and - 9
- 54
-6 and 9
- 54
Next, let's consider the coefficient of the linear term.
x^2+ 3x -54=0
For this term, we need the sum of the factors that produced the constant term to equal the coefficient of the linear term, 3.
Factors
Sum of Factors
1 and - 54
- 53
-1 and 54
53
2 and - 27
- 25
-2 and 27
25
3 and - 18
- 15
-3 and 18
15
6 and - 9
- 3
-6 and 9
3
We found the factors whose product is - 54 and whose sum is 3.
x^2+ 3x -54=0
⇕
(x-6)(x+9)=0
Solving the Equation
Now, as we have already factored the equation, we can apply the Zero Product Property to solve it.
