The Quadratic Formula

• What is the Quadratic Formula?
• Why Use the Quadratic Formula?
• Solving Equations using the Quadratic Formula

What is the Quadratic Formula

The quadratic formula is used to solve quadratic equations that are written in standard form and set equal to zero.  It uses the coefficients a, b, and c that are found in the standard equation.  The quadratic formula is:

It looks like a complete mess, doesn't it?  It can be a complicated formula to work through, but it is extremely useful in finding precise solutions to quadratic equations.  It is important to note that the equation must be in the form: 

Why Use the Quadratic Formula?

Perhaps one of the easiest ways to solve quadratic equations is factor the expression and use the Zero Factor Property.  For example:

However, consider this quadratic equation: 

This is known as a prime quadratic.  Prime quadratics cannot be written in factored form, which means that factoring and solving using the Zero Factor Property is out of the question.  In such cases, when we either cannot factor, or are having trouble figuring out how to factor it, we can use the quadratic formula.

The quadratic formula can also tell us if there is a real solution to the quadratic equation.  The expression underneath the radical is called the discriminant of the expression.  Since it is underneath a square root, it must have a non-negative value to result in a real number. 

• If the discriminant is non-negative (zero or greater), the quadratic has at least one real solution
• If the discriminant is negative (less than zero), the quadratic has no real solutions.

Solving Equations using the Quadratic Formula

Let's use the quadratic formula to solve the following equation: 

Don't jump the gun on identifying a, b, and c to use in the quadratic formula.  Remember, the equation needs to be set equal to zero before we begin to find solutions.  Always be sure to check that your equation is set equal to zero.  If it's not, we simply add or subtract terms from both sides of the equation, until zero is on one side of the equation, and everything else is on the other. 

An equivalent equation, and the equation we need to use to find a, b, and c, is: 

Here is how we use the quadratic formula to find solutions for x:

We used the equation set equal to zero to identify values to plug in for a, b, and c.  Now it is just a matter of performing operations in the correct order.  Here is how you should solve for x:

• Simplify what is underneath the radical first: evaluate b²​ and 4ac, then find the difference. This is called the discriminant.
• Add the square root of the discriminant to –b, and divide by 2a.  This is one solution for x.
• Subtract the square root of the discriminant from –b, and divide by 2a.  This is the other solution for x.

We both add and subtract the square root of the discriminant due to the ± symbol in the formula

Here are our worked out solutions: