How to Solve Quadratic Equations

There are three ways to learn how to solve quadratic equations. Choose your method or scroll down to view all three:

You may also be interested in our quadratic equations solver. You can find it for free at our quadratic formula page, or a larger version here.

An example of a quadratic equation and graph:

quadratic equation

Once your read our explanation, you will have no problem solving the equation at the right.

Let's get started:


What is a Quadratic Equation?

A quadratic equation has a term with x2 in it. See the examples below.

Quadratic Equation: x2 + 8x + 15 = 0 Not Quadratic: 5x = 15

The Standard Form of a Quadratic Equation

A quadratic equation is said to be in standard form whenever the exponents go in descending order. In other words, they fit the pattern...

  ax2 + bx + c = 0

Now that we have that out of the way, let's solve.

How to Solve Quadratic Equations by Factoring

Need more information about factoring? Factoring & Factoring Polynomials.

Once you have factored the polynomial, you will use the zero product property - basically you will set each factor = 0.

solve by factoring

To solve by factoring, you will first need to set the quadratic equal to 0. Then, you will factor the equation. For more information, click the links above.

After the equation is factored, set each individual factor equal to 0. Then solve the simple equations. Your final answers are x = 6, -7.

Solving Using the Quadratic Formula

Make sure to get your quadratic into standard form (see above), and then plug the values into the quadratic formula.

quadratic formula

Details of how to use the quadratic formula.


Solving by Completing the Square

Completing the square is the process of creating a perfect square trinomial.

To do this, you will add an equal amount to both sides of the equation. The amount that you add is determined by the b value.

First, convert the equation to the following form: ax2 + bx = c.

Next, you will add (b/2)2 to both sides of the equation. Finally, you will factore, take the square root of both sides, and then simplify the equation.


Take a look at the example and step by step instructions below of how to solve quadratic equations by completing the square.

Example: x2 + 6x = -2  

Add 9:


Square root:

Subtract 3:


Final Answer:

completing the square
Find the amount to add
complete the square  

Congratulations, you just learned three different methods of how to solve quadratic equations.

Return to more free algebra help or visit the GradeA homepage for more free math help.


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