Quadratic Equation Solver

A quadratic is a type of problem In mathematics that deals with a variable multiplied by itself (an operation known as squaring).

It derives from the area of a square being its side length multiplied by itself. The word quadratic comes from “quadratum”, which means square in Latin word.

There are a lot of phenomena in the real world that can be described by Quadratic equations. For example is rocket fly route, where it will land, how long it will take a person to row down & up a river, or how much to charge for a product.

Because of these applications, quadratics equations have profound historical importance and were foundation of algebra.

Quadratic Equation

Only if it can be put in the form ax2 + bx + c = 0, and a is not zero.

The name comes from “quad” meaning square, as the variable is squared (in other words x2).

These are all quadratic equations in disguise:

EquationIn standard forma, b and c
x2 = x + 1x² -x – 3 = 0a=1, b=-1, c=-3
2(x2 – 2x) = 22x² – 4x – 2 = 0a=2, b=-4, c=-2
x(x-6) = 18-x² +6x + 18 = 0a=-1, b=6, c=18
20 – 15/x – 10/x2 = 020x² -15x – 10 = 0a=20, b=-15, c=-10

Quadratic Formula

How Does this Work?

The solution(s) to a quadratic equation can be calculated using the Quadratic Formula:

Quadratic Formula

The “±” means we need to do a plus AND a minus, so there are normally TWO solutions !

The part (b2 – 4ac) is called the “discriminant“, because it can “discriminate” between the possible types of answer:

  • when b2 – 4ac > 0, we get two real solutions,
  • when b2 – 4ac = 0, we get just ONE solution,
  • when b2 – 4ac < 0, we get complex solutions.


Find x !

Equationa, b and cOperationx
x² -x – 3 = 0a=1, b=-1, c=-3Quadratic Formula Example 1Quadratic Formula Answer 1
2x² – 4x – 2 = 0a=2, b=-4, c=-2Quadratic Formula Example 2Quadratic Formula Answer 2

In order to Quadratic Formula to work, you must have your equation arranged in the form:

(quadratic) = 0

Also, the 2a in the denominator of the Formula is not just the square root. It is underneath everything above.

Make sure not to drop the square root or the “±” in the middle of your calculations, so you will not need to put them back in.

Remember that b2 means “the square of ALL of b, including its sign”. Do not leave b2 being negative, even if b is negative, because the square of a negative is a positive.

In other words, don’t be sloppy and don’t try to take shortcuts, because it will only hurt you in the long run. Trust me on this!

Learn More

Simplifying Square Roots

Cube Numbers and Cube Roots

Negative Exponents

Table of Factors and Multiples

Algebra Index