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Quadratic Formula Calculator


The quadratic formula calculator helps you to find the roots of the quadratic equation and shows step-by-step calculations. Our free online tool gives explicit results by optimizing the use of resources.

Quadratic Formula:

The Latin word “quadratic” comes from quadratum, generally used for square.

Our quadratic equation solver helps you to reduce the following second-degree expression to its roots:

$$ ax^2 + bx + c = 0 $$


  • x is the unknown value
  • a, b is the quadratic coefficient, a ≠ 0
  • c is a constant

Derivation of Quadratic Formula:

$$ \dfrac{a}{a}x^2 + \dfrac{b}{a}x + \dfrac{c}{a} = 0 $$

$$ x^2 + \dfrac{b}{a}x + \dfrac{c}{a} = 0 $$

$$ x^2 + \dfrac{b}{a}x = -\dfrac{c}{a} $$

Add $$ (\frac{b}{2a})^2 $$ on both sides of the equation.

$$ x^2 + \dfrac{b}{a}x + (\dfrac{b}{2a})^2 = -\dfrac{c}{a} + (\dfrac{b}{2a})^2 $$

$$ (x + \dfrac{b}{2a})^2 = -\dfrac{c}{a} + \dfrac{b^2}{4a^2} $$

$$ (x + \dfrac{b}{2a})^2 = \dfrac{b^2 – 4ac}{4a^2} $$

Take a square root on both sides of the equation.

$$ \sqrt{(x + \dfrac{b}{2a})^2} = \pm \sqrt{\dfrac{b^2 – 4ac}{4a^2}} $$

$$ x + \dfrac{b}{2a} = \pm \dfrac{\sqrt{b^2 – 4ac}}{2a} $$

$$ x = -\dfrac{b}{2a} \pm \dfrac{\sqrt{b^2 – 4ac}}{2a} $$

$$ x = \dfrac{-b \pm \sqrt{b^2 – 4ac}}{2a} $$

How To Use Quadratic Formula?

The quadratic formula calculator helps you to solve any quadratic equation in a matter of seconds. However, if your goal comes up with the manual calculations, you must delve into the example to explore more.


Suppose that we have 3x^2 - 5x + 2 = 0. We need to solve by the quadratic equation formula.


We already know about the formula that is:

$$ x = \dfrac{-b \pm \sqrt{b^2 – 4ac}}{2a} $$

And we also know the values of a, b, and c in the above-given statement:

a= 3, b= -5, and c= 2

Put the values in the formula. 

$$ x = \dfrac{-5 \pm \sqrt{-5^2 – 4(3)(2)}}{2(3)} $$

$$ x = \dfrac{5 \pm \sqrt{25 – 24)}}{6} $$

$$ x = \dfrac{5 \pm \sqrt{1}}{6} $$

Here the roots of a quadratic equation have two solutions. 

Root # 1

$$ x = \dfrac{5+1}{6} $$

$$ x =\dfrac{6}{6} $$

$$ x = 1 $$

Root # 2

$$ x = \dfrac{5-1}{6} $$

$$ x = \dfrac{4}{6} $$

$$ x = \dfrac{2}{3} $$

To learn more about quadratic equations and their calculations, find our complete quadratic tutorial for free.

Working of Quadratic Formula Calculator:

The quadratic root calculator will help you carry out calculations easily! Just enter the required values. Our quadratic function calculator will work best for you! 


  • Choose equation form
  • Choose computation method
  • Put the values of a, b, and c in the respective field
  • Press the “Calculate” button


  • Quadratic formula solver will give you an instant answer with the steps and graph shown


How Many Possibilities Does a Quadratic Equation Have?

There are three possibilities which include:

  • If the discriminant is positive, then there are 2 solutions 
  • If the discriminant is negative, then there is no solution 
  • If the discriminant equals 0, then there is 1 solution 

Can a Quadratic Formula be Used for Everything?

No, It is just used to solve the quadratic equations that are in the form of $$ ax^2 + bx + c = 0 $$. This form is also known as the standard form of the quadratic formula calculator.

How Many Roots Does a Quadratic Formula Equation Have?

A second-degree equation formula equation has two roots. It depends on the degree of the equation.


From the source Wikipedia: Quadratic equation, Solving the quadratic equation, Examples, and applications, History, Advanced topics

From the source Khan Academy: understanding quadratic formula, Example