Quadratic Equation definition: A quadratic equation in [katex]x[/katex] is an equation that can be written in the form of
[katex]a\;x^2-bx+c=0\;\;\;\;\;\;\;\;\;\;\;\;\;;a,\;b,\;c [/katex] are real numbers and [katex]a ≠ 0[/katex].
Another name for a quadratic equation in [katex]x[/katex] is 2nd Degree Polynomial in [katex]x[/katex].
Quadratic Equation Examples: The following equations are the quadratic equations:
- [katex]x^2-7x+9=0\;\;\;\;\;\;\;\;\;\;\;\;\;;a=1,\;b=-7,\;c=9 [/katex]
- [katex]5x^2+8x-9=0\;\;\;\;\;\;\;\;\;\;\;\;\;;a=5,\;b=8,\;c=-9 [/katex]
- [katex]-7 x^2-9x-8=0\;\;\;\;\;\;\;\;\;\;\;\;\;;a=-7,\;b=-9,\;c=-8 [/katex]
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
[katex]a\;x^2-bx+c=0\;\;\;\;\;\;\;\;\;\;\;\;\;;a,\;b,\;c [/katex] are real numbers and [katex]a ≠ 0[/katex].
It makes use of the fact that if [katex]xy = 0[/katex], then [katex]x = 0[/katex] or [katex]y = 0[/katex].
For Example:
If [katex](x – 2) (x – 4) = 0[/katex], then either [katex]x – 2 = 0[/katex] or [katex]x – 4 = 0[/katex]
Example 1: Solve the equation [katex]x^2-7x+10=0 [/katex] by factorization.
Solution:
[katex]x^2-7x+10=0 [/katex].
⇒[katex]x^2-5x-2x+10=0 [/katex]
⇒[katex]x(x-5)-2(x-5)=0 [/katex]
⇒ [katex](x – 2) (x – 5) = 0[/katex].
∴ either [katex]x – 2 = 0[/katex] ⇒ [katex]x = 2[/katex]
or [katex]x – 5 = 0[/katex] ⇒ [katex]x = 5[/katex]
∴ the given equation has two solutions: [katex]2[/katex] and [katex]5[/katex]
so Solution set = {2,5}