Quadratic Equation definition, Examples

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:

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.

quadratic equation

[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}

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