Question

If α and β are zeros of the quadratic polynomial x2-5x+k, such that α- β=1, find the value of k

Answer 1

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Answer 2

Given

$\alpha \: \: \: and \: \: \: \: \beta \: \: \:$

are the zeros of the quadratic polynomial

${x}^{2} - 5x + k$

$\alpha - \beta = 1$

Explanation

compare the given eqn with

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

therefor,

a = 1

b = -5

c = k

then,

$\alpha + \beta = \frac{ - b}{a}$

$= > \alpha + \beta = \frac{ - ( - 5)}{1}$

$\alpha + \beta = 5 .........(1)$

By the given condition,

$\alpha - \beta = 1 \\ \alpha = 1 + \beta$

put alpha = 1+ beta in eqn (1)

↪(1+beta)+beta =5

↪1+2beta= 5

↪2beta= 5-1

$\boxed{\textbf{\large{beta = 2}}}$

put( beta = 2) in eqn in( 1)

alpha + 2 = 5

alpha = 5-2

$\boxed{\textbf{\large{alpha= 3}}}$

we know ,

alpha X beta = c/a

= k/1

↪2 X 3 = k/1

↪6 = k

therefor the value of k is

$\boxed{\textbf{\large{k=6}}}$