Finding Zeros with Zero Trouble

By Ryan Branch

Zeros? Quadratic Functions? WHAT ARE THOSE???

There are many ways to find the zeros of Quadratic Functions. But first, we must understand what a quadratic function is, and what a zero is. A quadratic function is a function with a variable to the second degree. The variable is being squared, which is where the "quad" comes from in the term "quadratic function". A quadratic function can be given in standard form which is ax^2+bx+c. When graphed, quadratic functions make a parabola. These parabolas often cross the x-axis twice, but not always. The point at which the parabola touches the x-axis is called a zero, because this is the point(s) on the graph where the y value is 0. To "find the zeros" is also synonymous with "solving the quadratic" and often times "factoring".

3 Ways to Find the Zeros

There are 3 common ways to find the zeros. These ways are:

  • Factoring
  • Using Quadratic Formula
  • Completing the Square

Factoring Quadratic Functions: 3 Common Ways

Difference of 2 Squares

An 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.

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Completing the Square


If this sounds complicated, don't worry about it. It's actually a lot easier than it sounds. Remember how to solve a perfect square trinomial? Good! That is all this is. The point of this method is to be able to find the zeros when the function is not factorable by making the equation into a perfect square trinomial. Here is a sample equation: x^2+8x-4=0. This is not factorable so we need to make this a perfect square by adding the 4 over to the other side and then squaring half of "b" and adding that to both sides.


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

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Quadratic Formula

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Don't Judge a Book by its Cover

Ok, look, I get it. You probably took one look at this equation and wanted to vomit. Well please don't. It is actually so much simpler than it seems, so don't judge this equation by its cover. Remember standard form from earlier? That is all this is. Once you put the quadratic function into standard form, plug the terms in where they need to go and solve! This function will work for every quadratic trinomial.


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

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What should you know after reading this?

  1. What a Quadratic Function is
  2. What a zero is
  3. Standard form of a quadratic
  4. Difference of two squares
  5. Perfect square trinomial
  6. First degree polynomial factoring
  7. Completing the square
  8. Quadratic Formula

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