Roots of a Quadratic Equation

The solution of a quadratic equation, or the value/s obtained after solving a quadratic
equation are known as the roots of the quadratic equation. The roots of a quadratic
equation are known by the following names as well:

• Solutions of the quadratic equation
• Zeros of the quadratic equation
• x intercepts of the quadratic equation's graph

Mathematical definition:

Any real number 'alpha' is called a root/solution of a quadratic equation 'ax² + bx + c = 0'
if and only if

a(alpha)² + b(alpha) + c = 0 is true

In simple words, a number is called the root of a quadratic equation only if on
substituting the number in place of 'x' in the quadratic equation, the result is
true.

Example of a root of a quadratic equation:

For example, the, 5 is the root of the following quadratic equation because on putting x = 5 in it, we get LHS = RHS after simplification. That is,

Equation: x² - 2x - 15 = 0

Putting x = 5 in the above equation:

Put 5 in place of x in the equation:	52 - 2 * 5 - 10
Simplify the LHS:	25 - 10 - 15 = 0
	0 = 0

Thus we get 0 = 0 after simplifying the equation. This ensures that 5 is a root of the given quadratic equation.

In general, there are exactly two roots of a quadratic equation. To know why it is so, please go here:

Why does a quadratic equation have exactly two roots?

More examples :

• 4 and 3 are the roots of x² - 7x + 12 = 0
• 5 and 4 are the roots of x² - 9x + 20 = 0
• 2 and -1 are the roots of x² - x - 2 = 0