Math Doubts

Roots of a Quadratic function

Definition

The value of a variable that makes the quadratic function zero is called its root.

What is the Root of a Quadratic function?

A value can be substituted in the place of variable of a quadratic function to find its value. The quadratic function is equal to zero in some cases. So, the substituting value is called the root of a quadratic function. It is also called the zero of a quadratic function.

The highest exponent of a variable in a quadratic function is $2$. So, a quadratic function always has two roots (or zeros). So, let’s learn more about the roots of a quadratic function.

Example

$x^2-7x+12$

Substitute $x = 3$ to find its value.

$=\,\,$ $3^2-7(3)+12$

$=\,\,$ $3 \times 3-7 \times 3+12$

$=\,\,$ $9-21+12$

$=\,\,$ $0$

The value of quadratic function is equal to zero, when we substitute $x$ is equal to $3$. So, the number $3$ is called a root (or zero) of this quadratic function.

Similarly, substitute $x = 4$ to find the value of quadratic function.

$=\,\,$ $4^2-7(4)+12$

$=\,\,$ $4 \times 4-7 \times 4+12$

$=\,\,$ $16-28+12$

$=\,\,$ $0$

The value of quadratic function is also equal to zero, when we substitute $x$ is equal to $4$. So, the number $4$ is called a zero (or root) of this quadratic function.

It is proved that the quadratic function $x^2-7x+12$ is equal to zero only for $x = 3$ or $x = 4$. So, the numbers $3$ and $4$ are two roots of the quadratic function.

The above simple example helped you to understand what the roots of a quadratic function really are.

Actually, it is not always easy to guess the roots of a quadratic function. So, we require a special formula to find the roots of a quadratic function easily.

Formula to find the Roots

The roots of a quadratic function are denoted by alpha and beta in mathematics. The substitution of zeros $\alpha$ and $\beta$ in the place of variable makes quadratic function $ax^2+bx+c$ zero.

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

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

The following are the values of two solutions of a quadratic equation in general form.

$\therefore \,\,\,\,\,\,$ $\alpha \,=\, \dfrac{-b+\sqrt{b^2-4ac}}{2a}$ $\,\,\,$ or $\,\,\,$ $\beta \,=\, \dfrac{-b-\sqrt{b^2-4ac}}{2a}$

Now, they can be used as formulas to find the roots of a quadratic function in mathematics.

There are four methods to solve a quadratic equation in mathematics and let’s learn each method to know how to find the roots of a quadratic equation.