Quadratic Equation Calculator

26 Mar
Class 10, Class 10 - Chapter 4, Class 10 - Quadratic Equations, Math Concepts, , , ,
\( \begin{aligned} \end{aligned}\)

Quadratic Equation Calculator

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Enter the coefficeints of QE

Coeff. the x2

Coeff. the x

Constant

\( \begin{aligned}\text{Coeff of x}^2\text{ cannot be 0}\end{aligned}\)

Answer

Solution

Comparing the equation with
ax2 + bx + c = 0, we get

Discriminant (D)

Nature of Roots

D > 0 ⇒ Two Real and Distinct Roots

D = 0 ⇒ Two Real and Equal Roots

D < 0 ⇒ Two Complex Roots

Roots

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A polynomial equation of degree 2 is called Quadratic Equation. It’s standard form is

ax2 + bx + c = 0.

Roots of a Quadratic Equation

A quadratic equation has exactly 2 roots. These roots are either both Real or both Complex. It is not possible to have 1 real and 1 complex root for any quadratic equation with real coefficients. 

Click here to read about 1 real and 1 complex root scenario

Nature of Roots

To find the nature of roots, we need to find the discriminant of the quadratic equation. Formula to calculate the discriminant is:

\(\begin{aligned}&D = b^2-4ac\end{aligned}\)
D > 0 ⇒ Two Real and Distinct Roots

D = 0 ⇒ Two Real and Equal Roots

D < 0 ⇒ Two Complex Roots

It is always recommended to find the discriminant first rather than directly solving the equation.

Finding Roots

Once the discriminant is calculated, we can find the roots of the Quadratic Equation using the Quadratic Formula.

\(\begin{aligned}&x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\end{aligned}\)

An easy way to remember this formula is:

\(\begin{aligned}&x = \frac{-b \pm \sqrt{D}}{2a}\end{aligned}\)

Note: The roots of a quadratic equation are the zeros of equivalent quadratic polynomial. 

1 Real Root & 1 Complex Root Scenario

If we have a quadratic equation with imaginary or complex coefficients, then the equation can have 1 real and 1 complex root. For example, the roots of the equation mentioned below are: 3i and 1. 

\(\begin{aligned}&x^2-(3i + 1)x+(3i)=0\end{aligned}\)

An easy way to understand this is –

Form an equation like this:

\(\begin{aligned}&(x-3i) (x-1)=0\\\\&(x-\text{[A complex number]}) (x-\text{[A real number]})=0\end{aligned}\)