Deriving the Quadratic Formula
Start with the general quadratic equation:
\[
ax^2 + bx + c = 0
\]
Factor out \(a\):
\[
a\left(x^2 + \frac{b}{a}x + \frac{c}{a}\right) = 0
\]
Complete the square:
\[
x^2 + \frac{b}{a}x
= \left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a^2}
\]
So the equation becomes:
\[
a\left[\left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a^2} + \frac{c}{a}\right] = 0
\]
Remove the brackets:
\[
\left(x + \frac{b}{2a}\right)^2
= \frac{b^2}{4a^2} - \frac{c}{a}
\]
Simplify the right-hand side:
\[
\left(x + \frac{b}{2a}\right)^2
= \frac{b^2 - 4ac}{4a^2}
\]
Take the square root of both sides:
\[
x + \frac{b}{2a}
= \pm \frac{\sqrt{b^2 - 4ac}}{2a}
\]
Make \(x\) the subject:
\[
x = -\frac{b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a}
\]
Combine the fractions:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
