Question

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NayanShreyas21

question:-
If Alpha And beta are the zeroes of the polynomial f(x) = x2 - 5x-k such that alpha-beta= 1 find value of k​

Answer plz

Answer 1

Answer:

$\bold\red{k=-6}$

Step-by-step explanation:

It is being given that, alpha (α) and beta( β ) are the roots of the quadratic polynomial,

$f(x) = {x}^{2} - 5x - k$

Now, we know that, the standard form of a quadratic polynomial is,

$f(x) = a {x}^{2} + bx + c$

So according to this, comparing the coefficients of given polynomial, we get

a = 1

b = -5

c = -k

Also,

for a quadratic Polynomial,

the sum of roots, i.e.,

β + α  = -b/a

=> β + α = -(-5)/1 = 5 ............(i)

Also, it is being given that,

α - β = 1 .............(ii)

So, adding equation (i) and (ii), we get

2α = 6

=> α => 6/2 = 3

Therefore,

β = α - 1 = 3-1 = 2

Also, we know that, product of roots, i.e.,

β × α = c/a

=> 3 ×2 = -k/1

=> -k = 6

=> k = -6

Hence, the value of k = -6

Answer 2

${\bold{\underline{\underline{Answer:}}}}$

${\bold{\therefore k=-6}}$

${\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

• In the given question information given about alpha and beta are the zeroes of the polynomial f(x) = x2 - 5x-k such that alpha-beta= 1

• We have to find value of k.

$\underline \bold{Given : } \\ \implies \alpha \: and \: \beta \in ({x}^{2} - 5x - k = 0) \\ \\ \implies \alpha - \beta = 1\\ \\ \underline \bold{To \: Find : } \\ \implies Value \: of \: k = ?$

• According to given question :

$\bold{Sum \: of \: zeroes : } \\ \implies \alpha + \beta = - \frac{ b}{a} \\ \\ \implies \alpha + \beta = - \frac{ (- 5)}{1} \\ \\ \implies \alpha + \beta = 5 \\ \\ \implies \alpha = 5 - \beta - - - - - (1) \\ \\ \bold {Putting \: value \: of \: \alpha \: in \: given \: eqn} \\ \implies \alpha - \beta = 1 \\ \\ \implies 5 - \beta - \beta = 1 \\ \\ \implies - 2 \beta = 1 - 5 \\ \\ \implies \beta = \frac{ \cancel{- 4}}{ \cancel{- 2}} \\ \\ \implies \bold{\beta = 2} \\ \\ \bold{Putting \: value \: of \: \beta \: in \: (1)} \\ \implies \alpha = 5 - \beta \\ \\ \implies \alpha = 5 - 2 \\ \\ \bold{\implies \alpha = 3} \\ \\ \bold{Product \: of \: zeroes}\\ \implies \alpha \beta = \frac{c}{a} \\ \\ \implies 3 \times 2 = \frac{-k}{1} \\ \\ \bold{\implies k = -6}$