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How to Solve Quadratic Equations Using the Quadratic Formula (Mathematics Lesson)

What Is the Quadratic Formula?

The quadratic formula is a way of solving a quadratic equation.

For a quadratic equation:



The roots of the equation (the values of x that make y = 0) are given by the quadratic formula:



How to Solve a Quadratic Equation Using the Quadratic Equation

Question: Solve the quadratic equation below using the quadratic formula:



Step 1
Find the values of a, b and c.
a = 2, b = -5, c = 2.

Step 2
Insert the values of a, b and c into the quadratic formula and simplify:



Step 3
Find the root x1 that results from turning the ± into a +.



Step 4
Find the root x2 that results from turning the ± into a -.



The solution to the quadratic equation is x = 2 or x = ½.

A Real Example of How to Solve Quadratic Equations Using the Quadratic Formula

Sometimes quadratic equations have repeated roots - that is the same number solves the quadratic equation twice.

The slider below shows how to solve a quadratic equation where there are repeated roots.
Interactive Test
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2 ROOTS!

Quadratic equations have 2 roots, and the quadratic equation finds both of them.

Look closely at the formula, and you'll see a ± sign:



This means it is + one time, and - the other. This gives 2 roots:




Note

THE DISCRIMINANT

The term in the formula that appears in a square root is called the discriminant:



It discriminates between the 3 possible cases for the roots of a quadratic equation.

  • b2 - 4ac > 0 - there are 2 real, distinct roots.




  • b2 - 4ac = 0 - there is one repeated root.




  • b2 - 4ac < 0 - there are 2 complex roots.




  • BE CAREFUL WITH SIGNS

    a, b and c may be negative. Make sure you remember this when inserting them into the equation - write them inside brackets if need be.