Generative AI for Texts and Images

[Eli]

[Eli]

This is a proof of quadratic formulae generated by AI:

Proof:

Step 1: Complete the square

Factor out the coefficient of (x^2): $$a(x^2 + \frac{b}{a}x + \frac{c}{a}) = 0$$

Complete the square by adding and subtracting the square of half the coefficient of (x): $$\begin{align}a\left(x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2 + \frac{c}{a}\right) & = 0 \\ a\left(\left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a^2} + \frac{c}{a}\right) & = 0\end{align}$$

Step 2: Factor and solve

Factor the polynomial inside the parentheses: $$a\left(\left(x + \frac{b}{2a}\right)^2 - \frac{b^2 - 4ac}{4a^2}\right) = 0$$

Divide both sides by (a): $$\left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}$$

Take the square root of both sides: $$\left|x + \frac{b}{2a}\right| = \frac{\sqrt{b^2 - 4ac}}{2a}$$

Solve for (x): $$x = -\frac{b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a}$$

Step 3: Simplify

Combine the two solutions into a single expression: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$