Proof Formula ABC (Formula Quadaratic)

Abc formula (quadratic formula) is a formula that is used to find a solution of quadratic equations. This formula is usually used for the quadratic equation sticking with factoring. But to some, the formula is used as the primary method.

This formula is derived from the general form quadratic equations were solved by complete perfect square shape. So, before stepping into evidence abc formula.

This proves we start from the general form quadratic equation.

$ax^2 + bx + c = 0 \quad a \neq 0$

In order to convert it into squares, a value must be a number of squares. The simplest quadratic number is 1, so for both sides with a (division with a permissible, given $\, a \neq 0$ ).

$\frac{ax^2 + bx + c}{a} = \frac{0}{a}$
$x^2 + \tfrac{b}{a}x + \tfrac{c}{a} = 0$

Cut both sides with $\, \tfrac{c}{a}$ .

$x^2 + \tfrac{b}{a}x = - \tfrac{c}{a}$

Add $( \tfrac{b}{2a} )^2$ on both sides.

$x^2 + \tfrac{b}{a}x + ( \tfrac{b}{2a} )^2 = - \tfrac{c}{a} + ( \tfrac{b}{2a} )^2$

Change the left side will be a quadratic forms and simplify the right side.

$( x + \tfrac{b}{2a} )^2 = - \tfrac{c}{a} + \tfrac{b^2}{4a^2}$
$( x + \tfrac{b}{2a} )^2 = - \tfrac{4ac}{4a^2} + \tfrac{b^2}{4a^2}$
$( x + \tfrac{b}{2a} )^2 = \tfrac{b^2 - 4ac}{4a^2}$
$x + \tfrac{b}{2a} = \pm \sqrt{\tfrac{b^2 - 4ac}{4a^2}}$
$x = - \tfrac{b}{2a} \pm \sqrt{\tfrac{b^2 - 4ac}{4a^2}}$
$x = - \tfrac{b}{2a} \pm \tfrac{\sqrt{b^2 - 4ac}}{2a}$
$x = \tfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

This is the abc formula we often use in solving quadratic equations.

Source

Blog, Updated at: 07.59

Proof Formula ABC (Formula Quadaratic)

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