Quadratic Equation definition, Examples

3 Comments / rational and irrational numbers / By Azhar

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:

quadratic equation

$x^2-7x+10=0$ .

⇒ $x^2-5x-2x+10=0$

⇒ $x(x-5)-2(x-5)=0$
⇒ $(x - 2) (x - 5) = 0$ .

∴ 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}

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