Compare the steps needed to complete each task. How do both processes relate? Is one contained in the other?
See solution.
Practice makes perfect
There are many ways to solve a quadratic equation, one of them is by using the Zero Product Property when the equation is factored and set equal to zero. Let's consider the exercise's equation for example.
x^2-6x+8=0
We can solve this equation by following the next steps:
Factor the trinomial x^2-6x+8.
Use the Zero Product Property.
Solve the equivalent equations.As we can see, factoring the trinomial x^2-6x+8 is just a part of a larger process when solving the quadratic equation x^2-6x+8=0. Let's solve this equation by following the steps indicated above.
Factor the Trinomial x^2-6x+8
Note that the trinomial of interest is of the form x^2+bx+c.
x^2+ 2x+ c
x^2+( - 6)x+ 8
To factor it, we need to identify a pair of factors of c= 8, that add up to b= -6. Note that since we want their sum to be negative and their product to be positive, both factors must be negative.
Factors of 8
Sum
-8, -1
- 9 *
-4, -2
- 6 ✓
With the information shown above, we can factor the trinomial.
x^2-6x+8 ⇔ (x-4)(x-2)
Using the Zero Product Property
At this point the processes mentioned in the exercise deviate, the trinomial is already factored but the equation is still not solve. We have just rewritten the left-hand side.
x^2-6x+8=0
⇕
(x-4)(x-2)=0
However, since now the equation consists of a product that is set equal to zero we can make use of the Zero Product Property to find two equivalent equations.
(x-4)(x-2)=0
⇕
x-4 = 0 or x-2 = 0
Solving the Equivalent Equations
The solutions to the equivalent equations are the solutions to the original quadratic equation. We can solve each of them with one step.
x-&4 = 0 1cm x- -0.25cm&&2 = 0
&⇓ &&⇓
x&=4 && -0.2cmx=2
Therefore, the solutions for the quadratic equation, x^2-6x+8=0, are x=4 and x=2.
