How to Find the Roots of a Quadratic

By Kiran Gaunle
There's a formula that always works to find quadratic roots.

A quadratic equation, or a quadratic in short, is an equation in the form of ax^2 + bx + c = 0, where a is not equal to zero. The “roots” of the quadratic are the numbers that satisfy the quadratic equation. There are always two roots for any quadratic equation, although sometimes they may coincide.

You solve quadratic equations by completing the squares, factoring and by using the quadratic formula. However, since completing the squares and factoring are not universally applicable, it is best to learn and use the quadratic formula to find the roots of any quadratic equation.

The roots of any quadratic equation is given by: x = [-b +/- sqrt(-b^2 - 4ac)]/2a.

Write down the quadratic in the form of ax^2 + bx + c = 0. If the equation is in the form y = ax^2 + bx +c, simply replace the y with 0. This is done because the roots of the equation are the values where the y axis is equal to 0. For example, suppose the quadratic is 2x^2 - 20x + 5 = 0, where a = 2, b = -20, and c = 5.

Calculate the first root by using the formula x = [-b + sqrt(-b^2 - 4ac)]/2a. Substitute the values of a, b, and c. In our example, x = [20 + sqrt (20_20 - 4_2_5)]/2_5, which equals 9.7. Note that in order to find the first root, the first item inside the big brackets has changed its signs (because of double negative) and added to the second item.

Determine the second root by using the formula: x = [-b + sqrt(-b^2 - 4ac)]/2a. Note that the first item inside the big brackets is subtracted from the second to find the second root. In our example, x = [20 - sqrt(20_20 - 4_2_5)]/2_5, which equals 0.26.

Access the quadratic equation solver at Mathworld and enter the values of a, b, and c. Use this option if you do not want to use a calculator.


Negative numbers squared becomes positive. Make sure that you use correct signs.

About the Author

Kiran Gaunle is a freelancer based in New York. He started writing professionally in 2006. He has written research reports for the UN Development Programme and the "Kathmandu Post." Gaunle is working on a book of short stories and a novel. He holds a Master of Arts in international political economy and development from Fordham University.