Question

if alpha and beta are the zeroes of the polynomial f(X)=x2-8x+lambda, such that alpha-beta=2, find lambda​

Answer 1

GIVEN :–

• A quadratic equation f(x) = x² - 8x + λ have two roots α and β .

• α - β = 2 .

TO FIND :–

• Value of λ = ?

SOLUTION :–

• If a quadratic equation ax² + bx + c = 0 have two roots α and β , then α - β is –

$\\ \:\: \dashrightarrow \:\: { \boxed { \bold{ \alpha - \beta = \dfrac{ \pm \sqrt{d} }{ a} }}}$

• Here –

$\\ \:\: \blacktriangleright \:\:{ \bold{a = 1 }}$

$\\ \:\: \blacktriangleright \:\:{ \bold{b = - 8}}$

$\\ \:\: \blacktriangleright \:\:{ \bold{c = \lambda }}$

• Now put the values –

$\\ \implies{ \bold{ 2 = \dfrac{ \pm \sqrt{ {b}^{2} - 4ac } }{ a}}}$

$\\ \implies{ \bold{ 2 = \dfrac{ \pm \sqrt{ { (- 8)}^{2} - 4(1)( \lambda)}}{1}}}$

$\\ \implies{ \bold{ 2 = \pm \sqrt{{ (- 8)}^{2} - 4 \lambda}}}$

• Square on both sides –

$\\ \implies{ \bold{4 = {{ (- 8)}^{2} - 4 \lambda}}}$

$\\ \implies{ \bold{4 = 64 - 4 \lambda}}$

$\\ \implies{ \bold{4 \: \lambda= 64 - 4 }}$

$\\ \implies{ \bold{4 \: \lambda= 60}}$

$\\ \implies{ \bold{\: \lambda= \cancel \dfrac{60}{4}}}$

$\\ \implies \large{ \boxed{ \bold{ \lambda=15}}}$

Answer 2

Answer:

Step-by-step explanation:

${\red{\underline{\underline{\bold{Given:-}}}}}$

• are the zeroes of the quadratic equation f(x) = x² - 8x + λ

• α- β = 2

${\blue{\underline{\underline{\bold{To\:Find}}}}}$

• The value of λ

${\green{\underline{\underline{\bold{Solution:-}}}}}$

$\tt\pink{Formula \:used}$

$\boxed{\alpha - \beta = \frac{ \pm \sqrt{ {b}^{2} - 4ac } }{a}}$...........(1)

HEre:-

a = 1 ; b = -8 and c = λ

Substituting the values in equation(1)

$\implies 2 = \frac{\pm\sqrt{{(-8)}^{2} - 4(1)(\lambda)}}{1}$

$\implies 2 = \pm\sqrt{64- 4\lambda}$

Squaring on both sides :-

$\implies {2}^{2} = 64 - 4\lambda$

$\implies 4 = 64 - 4\lambda$

$\implies 4\lambda = 64-4$

$\implies 4\lambda = 60$

$\implies \lambda = \frac{60}{4}$

$\boxed{\lambda = 15}$