Sum and Product of Roots - Theory

Suppose that the quadratic equation

$$a{x^2} + bx + c = 0,\,\,\,a \ne 0$$

has the roots $\alpha$ and $\beta$. Thus, by the factor theorem, $\left( {x - \alpha } \right)$ and $\left( {x - \beta } \right)$ will be factors of the expression $a{x^2} + bx + c$:

$$a{x^2} + bx + c = a\left( {x - \alpha } \right)\left( {x - \beta } \right)$$

A factor of $a$ is required to equalize the coefficients on both sides. If we expand the right hand side, we have:

$$\begin{align}&a{x^2} + bx + c = a\left( {{x^2} - \left( {\alpha  + \beta } \right)x + \alpha \beta } \right)\\&\Rightarrow \,\,\,a{x^2} + bx + c = a{x^2} - a\left( {\alpha\,+\, \beta } \right)x + a\alpha \beta \end{align}$$

By comparing the coefficients on both sides, we obtain expressions for the sum and product of the roots  $\alpha$ and $\beta$:

$$\begin{align}&- a\left( {\alpha\, +\, \beta } \right) = b\,\,\, \Rightarrow \,\,\,S = \alpha\, + \,\beta\, =\, - \frac{b}{a}\\&a\,\alpha \beta\, =\, c\,\,\, \Rightarrow \,\,\,P = \alpha \beta\, = \,\frac{c}{a}\end{align}$$

Given a quadratic equation, we can evaluate the sum and product of its roots using these expressions. Here are two examples (we will use $\alpha$ and $\beta$ to denote the two roots):

$$\begin{align}&{x^2} - 3x + 2 = 0\\&\Rightarrow \,\,\,\left\{ \begin{array}{l}\alpha  + \beta  =  - \frac{b}{a} =  - \frac{{\left( { - 3} \right)}}{1} = 3\\\alpha \beta  = \frac{c}{a} = \frac{2}{1} = 2\end{array} \right.\\&2{x^2} - 5x + 2 = 0\\&\Rightarrow \,\,\,\left\{ \begin{array}{l}\alpha  + \beta  =  - \frac{b}{a} =  - \frac{{\left( { - 5} \right)}}{2} = \frac{5}{2}\\\alpha \beta  = \frac{c}{a} = \frac{2}{2} = 1\end{array} \right.\end{align}$$

We can also do the reverse: given the sum and product of the roots of a quadratic equation, we can obtain the equation. If the roots are $\alpha$ and $\beta$, we can write the equation as:

$$\begin{align}&\left( {x - \alpha } \right)\left( {x - \beta } \right) = 0\\&\Rightarrow \,\,\,{x^2} - \left( {\alpha  + \beta } \right)x + \alpha \beta  = 0\\&\Rightarrow \,\,\,{x^2} - Sx + P = 0\end{align}$$

For example, suppose that a quadratic equation is such that $S = 5$ and $P = 4$. This quadratic equation will be:

$$\begin{align}&{x^2} - Sx + P = 0\\&\Rightarrow \,\,\,{x^2} - 5x + 4 = 0\end{align}$$

As another example, suppose that $S = \frac{3}{2}$ and $P =  - \frac{1}{2}$ for a quadratic equation. This equation will be:

$$\begin{align}&{x^2} - Sx + P = 0\\&\Rightarrow \,\,\,{x^2} - \frac{3}{2}x - \frac{1}{2} = 0\\&\Rightarrow \,\,\,2{x^2} - 3x - 1 = 0\end{align}$$