Finding Zeros with Zero Trouble
By Ryan Branch
Zeros? Quadratic Functions? WHAT ARE THOSE???
3 Ways to Find the Zeros
- Using Quadratic Formula
- Completing the Square
Factoring Quadratic Functions: 3 Common Ways
Difference of 2 SquaresAn example of a difference of 2 squares is x^2-9=0. This quadratic function has two squares in. The first square is the x^2 term, and the second square is the 9. Because the 9 is being subtracted, it is a difference of two squares. To factor one of these is actually very simple.
1. Make sure the equation is in standard form (remember that from above?) and equal to 0. 2. Realize each term is a square, so set up two terms with the square root of "c" being added to the square root of "a", and one with the square root of "c" being subtracted from the square root of "c".
3. Set each factor to zero by itself, and solve for x.
Perfect Square Trinomial
An example of a perfect square trinomial is x^2-8x+16=0. Here, the quadratic function has three terms making it a trinomial. Hey, do you notice something different about this trinomial? Maybe something similar to the one we just learned how to factor? If you said "I see two squares in this!" then you are right. The first and last term of this trinomial are squares, and are very easy to solve as well. However, the middle term must be twice the value of "a" times "c" for it to be a Perfect Square trinomial.
1. Put the equation into standard form and make sure it is equal to 0.
2. Take the square of "a" and the square root of "c" and make two factors with them. Use the sign from "b" as the sign used to either add or subtract "c" from "a".
3. Set each factor equal to 0 and solve.
First Degree Trinomial
An example of a first degree trinomial is x^2+6x+8=0. Unfortunately this trinomial does not have a square at term "c" so we have to factor it a different way that is slightly more complicated. But don't be scared, we've made it this far haven't we? And we are still alive right? Ok, good. Let's look at these steps:
1. Put the equation into standard form and set it equal to zero
2. Look at the last term and the middle term
3. Ask yourself, "hmmm, what multiplies to "c" and adds to "b"?".
4. Put these numbers into a factor either added to or subtracted from x.
5. Set each factor equal to zero and solve.
Completing the Square
1. Add or subtract the current "c" value to the other side to get rid of it
2. Look at the "b" value, find half of it
3. Square that value (1/2b^2)
4. Add that value to both sides to make one side a perfect square trinomial
5. Find the square root of both sides
6. Solve for x
Don't Judge a Book by its Cover
1. Make sure the function is in standard form
2. Locate terms "a", "b", and "c" in the function
3. Substitute these terms in
4. Solve for x
What should you know after reading this?
- What a Quadratic Function is
- What a zero is
- Standard form of a quadratic
- Difference of two squares
- Perfect square trinomial
- First degree polynomial factoring
- Completing the square
- Quadratic Formula
- December 7/8 Quiz
- December 11 Unit 2 Test