Quadratic Equation definition: A quadratic equation in x is an equation that can be written in the form of
a\;x^2-bx+c=0\;\;\;\;\;\;\;\;\;\;\;\;\;;a,\;b,\;c are real numbers and a ≠0.
Another name for a quadratic equation in x is 2nd Degree Polynomial in x.
Quadratic Equation Examples: The following equations are the quadratic equations:
- x^2-7x+9=0\;\;\;\;\;\;\;\;\;\;\;\;\;;a=1,\;b=-7,\;c=9
- 5x^2+8x-9=0\;\;\;\;\;\;\;\;\;\;\;\;\;;a=5,\;b=8,\;c=-9
- -7 x^2-9x-8=0\;\;\;\;\;\;\;\;\;\;\;\;\;;a=-7,\;b=-9,\;c=-8
Solution of Quadratic Equations:
- There are three basic techniques for solving a quadratic equation:
- By Factorization.
- By Completing Squares, extracting square roots.
- By applying the Quadratic formula.
By Factorization: It involves factoring the polynomial
a\;x^2-bx+c=0\;\;\;\;\;\;\;\;\;\;\;\;\;;a,\;b,\;c are real numbers and a ≠0.
It makes use of the fact that if xy = 0, then x = 0 or y = 0.
For Example:
If (x - 2) (x - 4) = 0, then either x - 2 = 0 or x - 4 = 0
Example 1: Solve the equation x^2-7x+10=0 by factorization.
Solution:
x^2-7x+10=0 .
⇒x^2-5x-2x+10=0
⇒x(x-5)-2(x-5)=0
⇒ (x - 2) (x - 5) = 0.
∴ either x - 2 = 0 ⇒ x = 2
or x - 5 = 0 ⇒ x = 5
∴ the given equation has two solutions: 2 and 5
so Solution set = {2,5}