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Some of the content of this guide was modeled after a guide originally created by the Openstax and has been adapted for the GPRC Learning Commons in October 2020. This work is licensed under a Creative Commons BY 4.0 International License.

The Quadratic Formula


Another method to solve a quadratic equation is using the quadratic formula. For a quadratic equation of the form:


The quadratic formula is given by:


The number of solutions of a quadratic equation depends on the radical term in the quadratic formula. There are three cases for the solutions of a quadratic equation:


Case 1) , the equation has two solutions.
Case 2) , the equation has one solution.
Case 3) , the equation has no real solutions.


Example: Solve  using the quadratic formula.
We identify the following coefficients:

We now have to substitute the above values into the quadratic formula and simplify:


Example: Solve  using the quadratic formula.
We identify the coefficients in the quadratic equation:

Using the quadratic formula we get:


Example: Solve  using the quadratic formula. 
We have the following coefficients:

now, we put the above values in the quadratic formula:

since there is a negative number under the square root, the equation has no real solutions.



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