Quadratic formula

There are certain cases when solving a quadratic equation by factorisation or completing the square is time consuming, lengthy or difficult.

In such cases, we use the quadratic formula to solve the quadratic equation.


A quadratic equation is an equation of the form ax2 + bx + c=0 where a, b, and c are constants and a ≠ 0

For any such quadratic equation, the quadratic formula is:

Roots(x1, x2) = −b ± √   b2 − 4ac 
         2a          


Step by step process to solve the quadratic equation by quadratic formula:

Step I: Arrange the Quadratic Equation in the standard form of ax2 + b x + c = 0.

Step II: Compare the quadratic equation which is to be solved with the standard quadratic equation and find out the values of the coefficients a, b, c.

Step III: Put these values of a, b, c in Quadratic formula.

Roots(x1, x2) = −b ± √   b2 − 4ac 
         2a          


As you can notice in the formula. This implies, Quadratic formula calculates two values of x: x1 and x2, where

x1 = −b + √   b2 − 4ac 
         2a          

x2 = −b - √   b2 − 4ac 
         2a          

These two values of x for which ax2 + bx + c = 0 holds true, are called solutions of Quadratic Equation,or roots of Quadratic Equation.

Example: Solve, x2 - 10x = -24

Step I: - Convert the Quadratic Equation you wish to solve in Standard Form of Quadratic Equation, ax2 + bx + c = 0

x2 -10x = -24 is converted into x2 - 10x + 24 = 0

Step II: - Find value of coefficient a, b and c by comparing it with Standard Form of Quadratic Equation ax2 + bx + c = 0

Comparing x2 - 10x + 24 = 0 with ax2 + bx + c = 0, we get

a = 1,
b = -10,
c = 24

Step III: Putting values of a, b and c in the formula

Roots(x1, x2) = −b ± √   b2 − 4ac 
         2a          


x1 = −b + √   b2 − 4ac 
         2a          


=    −(-10) + √   (-10)2 − 4(1)(24)    = 6
         2(1)          


x2 = −b - √   b2 − 4ac 
         2a          


=    −(-10) - √   (-10)2 − 4(1)(24)    = 4
         2(1)          


6 and 4 are the roots of the quadratic equation x2 -10x = -24

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