Question

A journal on how to determine a quadratic equation given the roots ,or given the sum and the product of the roots

Answer 1

$</p><p>{\text{The quadratic equation }}a{x^2} + bx + c = 0\,{\text{can be formed using following cases}}{\text{.}} \hfill \\ {\text{Case}}\,{\text{I: }} \hfill \\ {\text{When distinct roots say }}\alpha \,{\text{and}}\,\beta \,{\text{is}}\,{\text{given, then the required quadratic equation will be}} \hfill \\ \left( {x - \alpha } \right)\left( {x - \beta } \right) = 0\, \hfill \\ {\text{Case}}\,{\text{II:}} \hfill \\</p><p> {\text{When sum of roots say }}S\,{\text{and}}\,{\text{product of roots }}P\,{\text{is}}\,{\text{given, then the required quadratic equation will be}} \hfill \\</p><p> {x^2} - \left( {{\text{sum of roots}}} \right)x + {\text{product of roots = }}0 \hfill \\</p><p> \Rightarrow \,{x^2} - Sx + P{\text{ = }}0 \hfill$

Answer 2

How to determine quadratic equation, if roots are given :- assume $\alpha$ and $\beta$ are the roots of a quadratic equation. then, write equation $(x-\alpha)(x-\beta)=0$ ,

if sum of roots and product of roots are given:
we know, quadratic equation is in the form of ax² + bx + c = 0, and we also know sum of roots = -b/a and product of roots = c/a
if you arrange ax² + bx + c = 0, you can see that x² - (-b/a)x + (c/a) = 0
e.g., x² - (sum of roots)x + product of roots = 0
then write the quadratic equation, x² - (sum of roots)x + product of roots = 0