Question

if α and β are roots of a quadratic equation, then what is α-β in terms of α+β and αβ.​

Answer 1

Step-by-step explanation:

We know that,

$\rm (\alpha - \beta)^2 = \alpha^2 +\beta^2 -2\alpha \beta$

and,

$\rm (\alpha +\beta)^2 = \alpha^2 +\beta^2 +2\alpha \beta$

$\rm (\alpha +\beta)^2-2\alpha \beta = \alpha^2 +\beta^2$

$\rm (\alpha -\beta)^2 = (\alpha +\beta)^2-2\alpha \beta -2\alpha \beta$

$\rm \alpha -\beta =\sqrt{ (\alpha +\beta)^2-4\alpha \beta }$

Answer 2

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

α and β are roots of a quadratic equation.

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

α-β in terms of α+β and αβ

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

Consider,

$\rm :\longmapsto\: {( \alpha - \beta )}^{2}$

$\rm \:  =  \: { \alpha }^{2} + { \beta }^{2} + 2 \alpha \beta$

can be rewritten as

$\rm \:  =  \: { \alpha }^{2} + { \beta }^{2} - 2 \alpha \beta + 2 \alpha \beta + 2 \alpha \beta$

$\rm \:  =  \: ({ \alpha }^{2} + { \beta }^{2} - 2 \alpha \beta) + 4\alpha \beta$

$\rm \:  =  \: {( \alpha - \beta) }^{2} + 4 \alpha \beta$

$\rm\implies \: {( \alpha - \beta) }^{2} = {( \alpha + \beta )}^{2} - 4 \alpha \beta$

$\rm\implies \: \alpha - \beta \: = \: \pm \: \sqrt{ {( \alpha + \beta )}^{2} - 4 \alpha \beta }$

Thus,

$\begin{gathered}\begin{gathered}\bf\: \alpha - \beta = \begin{cases} &\sf{ \sqrt{ {( \alpha + \beta )}^{2} - 4 \alpha \beta } \: \: when \: \alpha > \beta } \\ \\ &\sf{ - \sqrt{ {( \alpha + \beta )}^{2} - 4 \alpha \beta } \: \: when \: \alpha < \beta } \end{cases}\end{gathered}\end{gathered}$

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More Identities to know

(a + b)² = a² + 2ab + b²

(a - b)² = a² - 2ab + b²

a² - b² = (a + b)(a - b)

(a + b)² = (a - b)² + 4ab

(a - b)² = (a + b)² - 4ab

(a + b)² + (a - b)² = 2(a² + b²)

(a + b)³ = a³ + b³ + 3ab(a + b)

(a - b)³ = a³ - b³ - 3ab(a - b)