b We want to solve the given equation for x. To do this, we can start by using factoring. Then we will use the Zero Product Property.
Factoring
To factor a trinomial with a leading coefficient of 1, think of the process as multiplying two binomials in reverse. Let's start by taking a look at the constant term.
x^2+5x+6=0In this case, we have 6. 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 6
6
-1 and -6
6
2 and 3
6
-2 and -3
6
Next, let's consider the coefficient of the linear term.
x^2+5x+6=0
For this term, we need the sum of the factors that produced the constant term to equal the coefficient of the linear term, 5.
Factors
Sum of Factors
1 and 6
7
-1 and -6
-7
2 and 3
5
- 2 and - 3
- 5
We found the factors whose product is 6 and whose sum is 5.
x^2+5x+6=0 ⇔ (x+2)(x+3)=0
Zero Product Property
Since the equation is already written in factored form, we can now use the Zero Product Property.
d We want to solve the given equation for x. To do this, we can use factoring. We will start from identifying the greatest common factor (GCF). Then, we will use the Zero Product Property.
Factor Out The GCF
The GCF of an expression is a common factor of the terms in the expression. It is the common factor with the greatest coefficient and the greatest exponent. In this case, the GCF is x.
