The quadratic formula

We have seen that the general quadratic equation looks like $ax^2 + bx + c = 0$

We are going to use the technique of completing the square on the general equation to come up with a formula for solving any quadratic equation.

$ax^2 + bx + c = 0$

We divide by a

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

Then we complete the square

$(x + \frac{b}{2a})^2 - (\frac{b}{2a})^2 + \frac{c}{a} = 0$

Rearrange

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

Take the square root

$sqrt{(x + \frac{b}{2a})^2} = pmsqrt{(\frac{b}{2a})^2 - \frac{c}{a}}$

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

$x = -\frac{b}{2a}pmsqrt{(\frac{b}{2a})^2 - \frac{c}{a}}$

This is a formula that will find the roots of any quadratic equation but we can simplify it by multiplying the square root part by $\frac{2a}{2a}$ . This doesn’t change the result we get because $\frac{2a}{2a} = 1$ and multiplying by $1$ keeps the result the same.

When we put $2a$ inside the square root, we need to square it, so below you can see $\frac{sqrt{4a^2}}{2a}$ .

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

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

$x = -\frac{b}{2a}pm\frac{sqrt{{b^2} - 4ac}}{2a}$

$x = \frac{-bpmsqrt{{b^2} - 4ac}}{2a}$

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