UPSKILL MATH PLUS

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Consider the equation $$ax^2 + bx + c = 0$$, where $$a \ne 0$$.

The roots of the quadratic equation are $\frac{-b+\sqrt{{b}^{2}-4\mathit{ac}}}{2a}$ and $\frac{-b-\sqrt{{b}^{2}-4\mathit{ac}}}{2a}$.

If $$\alpha$$ and $$\beta$$ are the roots of a quadratic equation $$ax^2 + bx + c = 0$$, then:

$$\alpha =$$ $\frac{-b+\sqrt{{b}^{2}-4\mathit{ac}}}{2a}$ and $$\beta =$$ $\frac{-b-\sqrt{{b}^{2}-4\mathit{ac}}}{2a}$

Sum of the roots $$=$$ $$\alpha + \beta$$

$$=$$ $\frac{-b+\sqrt{{b}^{2}-4\mathit{ac}}}{2a}$ $$+$$ $\frac{-b-\sqrt{{b}^{2}-4\mathit{ac}}}{2a}$

$$=$$ $\frac{-b+\sqrt{{b}^{2}-4\mathit{ac}}-b-\sqrt{{b}^{2}-4\mathit{ac}}}{2a}$

$$=$$ $\frac{-2b}{2a}=\frac{-b}{a}$
Sum of the roots $$=$$ $$\alpha + \beta$$ $$=$$ $\frac{-b}{a}$
Product of the roots $$=$$ $$\alpha \beta$$

$$=$$ $\frac{-b+\sqrt{{b}^{2}-4\mathit{ac}}}{2a}$ $$\times$$ $\frac{-b-\sqrt{{b}^{2}-4\mathit{ac}}}{2a}$

$$=$$ $\frac{\left(-b\right)\left(-b\right)+\left(-b\right)\left(-\sqrt{{b}^{2}-4\mathit{ac}}\right)+\left(\sqrt{{b}^{2}-4\mathit{ac}}\right)\left(-b\right)+\left(\sqrt{{b}^{2}-4\mathit{ac}}\right)\left(-\sqrt{{b}^{2}-4\mathit{ac}}\right)}{2a×2a}$

$$=$$ $\frac{{b}^{2}+b\sqrt{{b}^{2}-4\mathit{ac}}-b\sqrt{{b}^{2}-4\mathit{ac}}-\left({b}^{2}-4\mathit{ac}\right)}{4{a}^{2}}$

$$=$$ $\frac{{b}^{2}-{b}^{2}+4\mathit{ac}}{4{a}^{2}}$

$$=$$ $\frac{4\mathit{ac}}{4{a}^{2}}=\frac{c}{a}$
Product of the roots $$=$$ $$\alpha \beta$$ $$=$$ $\frac{c}{a}$
Since $$(x - \alpha)$$ and $$(x - \beta)$$ are factors of $$ax^2 + bx + c = 0$$:

$$(x - \alpha) (x - \beta) = 0$$

$$\Rightarrow x^2 - \alpha x - \beta x + \alpha \beta = 0$$

$$\Rightarrow x^2 - (\alpha + \beta) x + \alpha \beta = 0$$

$$\Rightarrow x^2 - (\text{sum of roots}) x + \text{product of roots} = 0$$
If $$\alpha$$ and $$\beta$$ are the roots of a quadratic equation, then the general formula to construct the quadratic equation is $$x^2 - (\alpha + \beta) x + \alpha \beta = 0$$.

That is, $$x^2 - (\text{sum of roots}) x + \text{product of roots} = 0$$.