Quadratic equation:
A quadratic equation in the variable \(x\) is an equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\) and \(c\) are numbers, \(a \ne 0\). The degree of the quadratic equation is \(2\).
Roots of a quadratic equation:
The value of \(x\) that makes the expression \(ax^2 + bx + c\) is zero, called the roots of the quadratic equation.
Solution of the quadratic equation:
The solution of the quadratic equation is the value of the variable that makes the equation zero. We can say these solutions as roots or zeroes.
There are three methods to solve the quadratic equations.
- Factorisation
- Completing the square
- Quadratic formula
Solving a quadratic equation by using quadratic formula:
Let us see the procedure to solve the quadratic equation by using quadratic formula.
Step - 1: Compare the given equation with the standard form of quadratic equation \(ax^2 + bx + c=0\) and obtain the values of \(a\), \(b\) and \(c\).
Step - 2: Substitute the values of \(a\), \(b\) and \(c\) in the quadratic formula and solve for \(x\).
The formula for finding the roots of the quadratic equation \(ax^2 + bx + c = 0\) is:
This formula is known as the quadratic formula.
Example:
1. Find the roots of \(2x^2 + 3x - 77 = 0\) by using quadratic formula.
Solution:
The given equation is \(2x^2 + 3x - 77 = 0\).
Here, \(a = 2\), \(b = 3\) and \(c = -77\).
Quadratic formula:
Substitute the given values in the formula.
or
or
\(x =\) or \(x = -7\)
Therefore, the roots of the given equation are \(-7\) and .