How Euler Might Prove the Quadratic Formula

Today, I'm going to show you a completely unique way to come up with the quadratic formula using complex numbers. We start by substituting the polar form of a complex number, $x = r e^{i\theta}$, into the standard quadratic equation $ax^2 + bx + c = 0$. By using Euler's formula and De Moivre's theorem, we expand the equation and organize it into its real and imaginary parts.

Because the entire equation equals zero, both the real and imaginary parts must individually equal zero, giving us a system of two equations. Using trigonometric tools like double angle identities and the Pythagorean identity, we isolate our target expressions: $r\cos(\theta)$ and $r\sin(\theta)$. After combining the terms and a bit of algebraic manipulation, we end up with a strange-looking formula featuring an $i$ and a reversed discriminant ($4ac - b^2$). Finally, by substituting $i$ as $\sqrt{-1}$ and multiplying it inside the square root, we flip the signs to perfectly arrive at the classic quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$!
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math, mathematics, algebra, quadratic formula, Euler's formula, Euler's identity, complex numbers, polar form, De Moivre's theorem, math proof, quadratic equation derivation, double angle identity, Pythagorean identity, blackpenredpen math, advanced algebra.

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