1. The "third method" often refers to solving quadratic equations by completing the square. 2. Problem: Solve $ax^2 + bx + c = 0$ using completing the square. 3. Formula and rules: - Rewrite as $a(x - h)^2 = k$ where $h,k$ are constants. - This allows solving by square roots. 4. Steps: - Start with $ax^2 + bx + c = 0$. - Divide both sides by $a$ (assuming $a \neq 0$): $$x^2 + \frac{b}{a}x + \frac{c}{a} = 0$$ - Move constant term: $$x^2 + \frac{b}{a}x = -\frac{c}{a}$$ - Add $\left(\frac{b}{2a}\right)^2$ to both sides: $$x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2$$ - Left side is a perfect square: $$\left(x + \frac{b}{2a}\right)^2 = -\frac{c}{a} + \frac{b^2}{4a^2}$$ - Simplify right side: $$\left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}$$ - Take square root: $$x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2a}$$ - Solve for $x$: $$x = -\frac{b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a}$$ 5. This is the quadratic formula derived by completing the square. 6. Notes: - Completing the square transforms the quadratic to an easy-to-solve form. - The discriminant $b^2 - 4ac$ determines root nature.