Roots of a Quadratic Equation

The values of variable for which the quadratic expression becomes zero are called the roots of a quadratic equation.

Introduction

The highest exponent of a variable in a term of any quadratic expression is two. Hence, the value of every quadratic expression is equal to zero for two suitable values of variable. The two solutions are called the roots of quadratic equation.

A quadratic expression is written algebraically as $ax^2+bx+c$ in mathematics.

In this standard form quadratic expression, the highest power of variable is two. Therefore, this algebraic expression should be equal to zero for two suitable values.

If the two suitable values of variable $x$ are denoted by $\alpha$ and $\beta$, then the quadratic expression becomes zero when we substitute each value in the place of variable $x$ in quadratic expression.

$(1).\,\,\,$ $a(\alpha)^2+b(\alpha)+c$ $\,=\,$ $0$

$(2).\,\,\,$ $a(\beta)^2+b(\beta)+c$ $\,=\,$ $0$

In this case, the alpha and beta are called the roots of quadratic equation, and they can be evaluated by the following formula.

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

Example

$x^2-7x+12 \,=\, 0$

It is an example quadratic equation and this equation is equal to zero for two values. Substitute $x = 3$ and $x = 4$ in the quadratic expression $x^2-3x+2$ and see the result.

$(1).\,\,\,$ $(3)^2-7(3)+12$ $\,=\,$ $9-21+12$ $\,=\,$ $0$

$(2).\,\,\,$ $(4)^2-7(4)+12$ $\,=\,$ $16-28+12$ $\,=\,$ $0$

The quadratic expression $x^2-3x+2$ is equal to zero for $x = 3$ and $x = 4$. Hence, the values $3$ and $4$ are called the roots of quadratic equation $x^2-7x+12 \,=\, 0$

Methods

There are four methods to evaluate the roots of any quadratic function in mathematics. So, let’s learn how to find the roots of quadratic equation with understandable examples.

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