Explanation vs Derivation Math Example 1

Solution

1. 1
   Derivation — start from $ax^2+bx+c=0$ and complete the square. Divide by $a$: $x^2 + \frac{b}{a}x = -\frac{c}{a}$. Add $\left(\frac{b}{2a}\right)^2$ to both sides: $\left(x+\frac{b}{2a}\right)^2 = \frac{b^2-4ac}{4a^2}$.
2. 2
   Take square roots of both sides: $x + \frac{b}{2a} = \pm\frac{\sqrt{b^2-4ac}}{2a}$. Solve for $x$: $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$.
3. 3
   Conceptual explanation: the formula locates each root at signed distance $\frac{\sqrt{b^2-4ac}}{2a}$ from the axis of symmetry $x = -\frac{b}{2a}$. The discriminant $\Delta = b^2-4ac$ signals: $\Delta > 0$ gives two distinct real roots, $\Delta = 0$ gives one repeated root, $\Delta < 0$ gives two complex-conjugate roots.