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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:


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

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