1. Let's start by stating the problem: We want to find the sum and product of the roots of a quadratic equation and express the roots $\alpha$ and $\beta$ as the subject. 2. Consider the general quadratic equation: $$ax^2 + bx + c = 0$$ where $a \neq 0$. 3. The roots of this quadratic equation are $\alpha$ and $\beta$. 4. According to Vieta's formulas, the sum and product of the roots are given by: - Sum of roots: $$\alpha + \beta = -\frac{b}{a}$$ - Product of roots: $$\alpha \beta = \frac{c}{a}$$ 5. To make $\alpha$ and $\beta$ the subject, we use the fact that they satisfy the quadratic equation: $$x^2 - (\alpha + \beta)x + \alpha \beta = 0$$ Substituting the sum and product, we get: $$x^2 - \left(-\frac{b}{a}\right)x + \frac{c}{a} = 0$$ which simplifies to: $$x^2 + \frac{b}{a}x + \frac{c}{a} = 0$$ 6. Solving for $x$ (which represents $\alpha$ or $\beta$) using the quadratic formula: $$x = \frac{-\frac{b}{a} \pm \sqrt{\left(\frac{b}{a}\right)^2 - 4 \cdot 1 \cdot \frac{c}{a}}}{2}$$ 7. Simplify the expression: $$x = \frac{-\frac{b}{a} \pm \sqrt{\frac{b^2}{a^2} - \frac{4c}{a}}}{2} = \frac{-\frac{b}{a} \pm \frac{\sqrt{b^2 - 4ac}}{a}}{2}$$ 8. Multiply numerator and denominator by $a$ to clear denominators: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 9. Therefore, the roots $\alpha$ and $\beta$ are: $$\alpha, \beta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ This shows how to find the sum and product of roots and express the roots themselves as the subject of the quadratic equation.