1. The "third method" usually refers to a specific approach in solving a problem, but since you didn't specify the problem, I'll explain a common third method in algebra: solving quadratic equations by completing the square. 2. The problem: Solve a quadratic equation of the form $$ax^2 + bx + c = 0$$ using the third method: completing the square. 3. Formula and rules: - We want to rewrite the quadratic in the form $$a(x - h)^2 = k$$ where $$h$$ and $$k$$ are constants. - This method helps us find the roots by taking the square root of both sides. 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 the constant term to the right side: $$x^2 + \frac{b}{a}x = -\frac{c}{a}$$. - To complete the square, add $$\left(\frac{b}{2a}\right)^2$$ to both sides: $$x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2$$. - The left side is now a perfect square: $$\left(x + \frac{b}{2a}\right)^2 = -\frac{c}{a} + \frac{b^2}{4a^2}$$. - Simplify the right side: $$\left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}$$. - Take the square root of both sides: $$x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2a}$$. - Finally, 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, showing how the third method works. 6. Important notes: - Completing the square transforms the quadratic into a form that is easy to solve. - The discriminant $$b^2 - 4ac$$ determines the nature of the roots. This explanation covers the third method properly with all intermediate steps and reasoning.