Question

If -1 and 3 are the roots of x^2 + px + q = 0, find the values of p and q.

Answer 1

$\large\underline{\sf{Given- }}$

$\rm :\longmapsto\: - 1 \: and \: 3 \: are \: the \: roots \: of \: {x}^{2} + px + q = 0$

$\large\underline{\sf{To\:Find - }}$

$\boxed{ \bf{ \: p \: \: and \: \: q}}$

$\large\underline{\sf{Solution-}}$

Given that

$\rm :\longmapsto\: - 1 \: and \: 3 \: are \: the \: roots \: of \: {x}^{2} + px + q = 0$

We know that

$\boxed{\red{\sf Product\ of\ the\ roots=\frac{Constant}{coefficient\ of\ x^{2}}}}$

$\rm\implies\: - 1 \times 3 = \dfrac{q}{1}$

$\bf\implies \:q = \: - \: 3$

Also,

$\boxed{\red{\sf Sum\ of\ the\ roots=\frac{-coefficient\ of\ x}{coefficient\ of\ x^{2}}}}$

$\rm\implies\: - 1 + 3 \: = - \: \dfrac{p}{1}$

$\bf\implies \:p \: = \: - \: 2$

$\begin{gathered}\begin{gathered}\rm :\longmapsto\:\bf\: Hence \: - \: \begin{cases} &\bf{p \: = \: - \: 2} \\ &\bf{q \: = \: - \: 3} \end{cases}\end{gathered}\end{gathered}$

Additional Information :-

$\red{\rm :\longmapsto\: \alpha , \beta , \gamma \: are \: roots \: of \: a {x}^{3} + b {x}^{2} + cx + d = 0, \: then}$

$\boxed{ \bf{ \: \alpha + \beta + \gamma = - \dfrac{b}{a}}}$

$\boxed{ \bf{ \: \alpha \beta + \beta \gamma + \gamma \alpha = \dfrac{c}{a}}}$

$\boxed{ \bf{ \: \alpha \beta \gamma = - \dfrac{d}{a}}}$