Question

if alpha and beta are zeros of given polynomial f x is equals to x square - 3 bracket X + 2 bracket minus C then find the value of alpha + 2 and beta + 2
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Answer 1

Answer:

Step-by-step explanation:

Given :

α and β are the zeroes of the polynomial,

f(x) = x² - 3(x + 2) - c,. f(x) = 0,

To Find :

The value of (α + 2) and (β + 2)

Solution :

We know that, quadratic equations of the form,

ax² + bx + c = 0

have zeroes α and β in the form,

x² - (α + β) x + αβ = 0,.

⇒$\alpha +\beta =\frac{-b}{a}$

⇒$\alpha\beta= \frac{c}{a}$

Hence,

By expanding the given equation,

x² - 3(x + 2) - c = 0

x² - 3x - 6 - c = 0

Here,

a = 1 , b = -3 , c = - 6 - c

Hence,

by applying the formula,

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

$\alpha\beta=\frac{c}{a}$

⇒ $\alpha+\beta=\frac{-(-3)}{1} = 3$

⇒ α + β = 3 ....(1)

⇒ $\alpha\beta=\frac{-6-c}{1}$ = - 6 - c = - (c + 6)

⇒ αβ = - ( c + 6 )

By solving we get,

α = $\frac{3 +\sqrt{4c + 33} }{2}$

β = $\frac{3 -\sqrt{4c + 33} }{2}$

⇒ α + 2 = $\frac{3 +\sqrt{4c + 33} }{2} + 2 = \frac{7 +\sqrt{4c + 33} }{2}$

⇒ β + 2 =  $\frac{3 -\sqrt{4c + 33} }{2} + 2 = \frac{7 -\sqrt{4c + 33} }{2}$