Quadratic formula

How to derive quadratic formula ?

Quadratic Equation:

[latex s=3]x = \frac{{ – b \pm \sqrt {b^2 – 4ac} }}{{2a}}[/latex]

Now we will see how to make that formula. Basically quadratic equation is a second order polynomial with 3 coefficients. The quadratic equation is given by:

[latex s=2]ax^2+bx+c=0[/latex]

Divide the equation with ‘a’

[latex s=3]\frac{ax^2}{a}+\frac{bx}{a}+\frac{c}{a}=0[/latex]

[latex s=3]x^2+\frac{bx}{a}+\frac{c}{a}=0[/latex]

[latex s=3]x^2+\frac{bx}{a}=-\frac{c}{a}[/latex]

Now add       [latex s=2](\frac{b}{2a})^2[/latex]       on both side. The reason to add is that we want to make a complete square on left hand side.

[latex s=3]x^2+\frac{bx}{a}+(\frac{b}{2a})^2=-\frac{c}{a}+(\frac{b}{2a})^2[/latex]

Now take Left Hand Side and make its complete square.

[latex s=3]x^2+\frac{bx}{a}+(\frac{b}{2a})^2[/latex]

Let’s suppose     [latex s=2]t=\frac{b}{2a}[/latex]      so the Left Hand Equation become

[latex s=2]x^2+2tx+t^2[/latex]

[latex s=2](x+t)^2[/latex]

Now replace ‘t’ with     [latex s=2]\frac{b}{2a}[/latex]

[latex s=2](x+\frac{b}{2a})^2[/latex]

and our equation will look like

[latex s=3](x+\frac{b}{2a})^2=-\frac{c}{a}+(\frac{b}{2a})^2[/latex]

Now we have to rearrange the equation and solve for ‘x’. Take square root on both sides of above equation.

[latex s=3]\sqrt{(x+\frac{b}{2a})^2}=\pm \sqrt{-\frac{c}{a}+(\frac{b}{2a})^2}[/latex]

[latex s=3]x+\frac{b}{2a}=\pm \sqrt{-\frac{c}{a}+(\frac{b}{2a})^2}[/latex]

Move      [latex s=2]\frac{b}{2a}[/latex]     to right side

[latex s=3]x=-\frac{b}{2a} \pm \sqrt{-\frac{c}{a}+(\frac{b}{2a})^2}[/latex]

Simplify the equation. Multiply right by     [latex s=2]\frac{2a}{2a}[/latex]

[latex s=3]\frac{-b \pm \sqrt{-(\frac{c}{a}) \times (2a)^2+(\frac{b}{2a})^2 \times (2a)^2}}{2a}[/latex]

So we made it. The Quadratic formula is

[latex s=3]x = \frac{{ – b \pm \sqrt {b^2 – 4ac} }}{{2a}}[/latex]

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