2.5 Formation of a quadratic equation

2.5 Formation of a quadratic equation.

If $\alpha$$\alpha$ and $\beta$$\beta$ are the roots of the required quadratic equation. Let $x=\alpha$$x = \alpha$ and $x=\beta$$x = \beta$ i.e., $x-\alpha =0$$x - \alpha = 0$ $x-\beta =0$$x - \beta = 0$ and $(x-\alpha )(x-\beta )=0$$(x - \alpha)(x - \beta) = 0$ $x^2-(\alpha +\beta )x+\alpha \beta =0$$x^{2} - (\alpha +\beta)x + \alpha \beta = 0$ which is the required quadratic equation in the standard form.

2.5.1 Find a quadratic equation from given roots and establish the formula $x^{2} - (\text{sum of the roots})x + \text{product of the roots} = 0$

Let $\alpha ,\beta$$\alpha ,\beta$ be the roots of the quadratic equation $ax^{2} + bx + c = 0$ (a $\neq 0$ (i)

Then $\alpha +\beta =-\frac{b}{a}$$\alpha +\beta = -\frac{b}{a}$ and $\alpha \beta =\frac{c}{a}$$\alpha \beta = \frac{c}{a}$

Rewrite eq. (i) as $x^2+\frac{b}{a}x+\frac{c}{a}=0$$x^{2} + \frac{b}{a} x + \frac{c}{a} = 0$ or $x^2-(-\frac{b}{a})x+\frac{c}{a}=0$$x^{2} - \left(-\frac{b}{a}\right)x + \frac{c}{a} = 0$ $x^2-(\alpha +\beta )x+\alpha \beta =0$$x^{2} - (\alpha +\beta)x + \alpha \beta = 0$ or $x^{2} - (\text{sum of roots})x + \text{product of roots} = 0$ , that is, $x^{2} - Sx + P = 0$ where $S=\alpha +\beta$$S = \alpha +\beta$ and $P=\alpha \beta$$P = \alpha \beta$