Let's start by rewriting the equation in such way that all the terms are on the left side of the equality sign.
22x-x^2=96
⇕
22x-x^2-96=0
To make the process of factoring easier, let's rearrange the order of the terms in such way that they follow the standard form of a quadratic function.
22x-x^2-96=0
⇕
- x^2+22x-96=0
To completely factor the given expression, we will now rewrite the expression as a trinomial with a leading coefficient of 1.
- x^2+22x-96 = - (x^2-22x+96)
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^2-22x+ 96)=0
In this case, we have 96. This is a positive number, so for the product of the constant terms in the factors to be positive, these constants must have the same sign (both positive or both negative.)
Factor Constants
Product of Constants
1 and 96
96
-1 and - 96
96
2 and 48
96
-2 and - 48
96
3 and 32
96
-3 and - 32
96
4 and 24
96
-4 and - 24
96
6 and 16
96
-6 and - 16
96
8 and 12
96
-8 and - 12
96
Next, let's consider the coefficient of the linear term.
-(x^2 -22x+ 96)=0
For this term, we need the sum of the factors that produced the constant term to equal the coefficient of the linear term, - 22.
Factors
Sum of Factors
1 and 96
97
-1 and - 96
- 97
2 and 48
50
-2 and - 48
- 50
3 and 32
35
-3 and - 32
- 35
4 and 24
28
-4 and - 24
- 28
6 and 16
22
-6 and - 16
- 22
8 and 12
20
-8 and - 12
- 20
We found the factors whose product is 96 and whose sum is - 22.
- (x^2 -22x+ 96)=0
⇕
- (x-6)(x-16)=0
Solving the Equation
Now, as we have already factored the equation, we can apply the Zero Product Property to solve it.
