Quadratic Equation Derivation:
$$
ax^2 + bx + c = 0
$$
Divide through by \(a\):
$$
x^2 + \frac{b}{a}x + \frac{c}{a} = 0
$$
Move the constant term:
$$
x^2 + \frac{b}{a}x = -\frac{c}{a}
$$
Complete the square:
$$
x^2 + \frac{b}{a}x + \left( \frac{b}{2a} \right)^2 = \left( \frac{b}{2a} \right)^2 – \frac{c}{a}
$$
Factor:
$$
\left( x + \frac{b}{2a} \right)^2 = \frac{b^2}{4a^2} – \frac{c}{a}
$$
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:
$$
x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 – 4ac}}{2a}
$$
Isolate \(x\):
$$
x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}
$$
