Question

If a and b zeroes of polynomial x²- x-k, such that a-b =9 find K.
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Answer 1

Solution

Given :-

• Polynomial, x² - x - k
• a & b are zeroes
• a-b =9_____(1)

Find :-

• Value of k

Step - by - Step - Explanation

Using Formula,

★ Sum of zeroes = -(coefficient of x)/(coefficient of x²)

★ Product of zeroes = (constant part)/(coefficient of x²)

_________________________

Now,

➡ Sum of Zeroes = -(-1)/1

➡a + b = 1 ____________(2)

And,

➡Product of zeroes = -k

➡ ab = -k _____________(3)

Add equ(1) & equ(2)

➡ 2a = = 9 + 1

➡2a = 10

➡ a = 10/2

➡ a = 5

Keep value of a in equ(2)

➡ 5 + b = 1

➡ b = 1 - 5

➡ b = -4

____________

Now, keep value of a & b in equ(3)

➡ 5 × -4 = -k

➡-k = -20

➡k = 20

Hence

• Value of k will be = 20

_________________

Answer 2

Solution :

$\underline{\bf{Given\::}}}$

We have quadratic polynomial p(x) = x² - x - k & zero of the polynomial p(x) = 0.

⇒ α-β = 9..............(1)

$\underline{\bf{Explanation\::}}}$

As we know that given polynomial compared with ax² + bx + c;

• a = 1
• b = -1
• c = -k

Now;

$\underline{\mathcal{SUM\:OF\:THE\:ZEROES\::}}$

$\mapsto\sf{\alpha +\beta=\dfrac{-b}{a} =\bigg\lgroup \dfrac{Coefficient\:of\:x}{Coefficient\:of\:x^{2}}\bigg\rgroup}\\\\\\\mapsto\sf{\alpha +\beta =\dfrac{-(-1)}{1} }\\\\\mapsto\sf{\alpha +\beta =\dfrac{1}{1}}\\\\\mapsto\sf{\alpha +\beta =1}\\\\\mapsto\sf{\alpha =1 - \beta ......................(2)}$

Putting the value of α in equation (1),we get;

$\mapsto\sf{1-\beta -\beta =9}\\\\\mapsto\sf{1-2\beta =9}\\\\\mapsto\sf{-2\beta =9-1}\\\\\mapsto\sf{-2\beta =8}\\\\\mapsto\sf{\beta =-\cancel{8/2}}\\\\\mapsto\bf{\beta =-4}$

Putting the value of β in equation (2),we get;

$\mapsto\sf{\alpha =1-(-4)}\\\\\mapsto\sf{\alpha =1+4}\\\\\mapsto\bf{\alpha =5}$

$\underline{\mathcal{PRODUCT\:OF\:THE\:ZEROES\::}}$

$\mapsto\sf{\alpha \times \beta=\dfrac{c}{a} =\bigg\lgroup \dfrac{Constant\:term}{Coefficient\:of\:x^{2}}\bigg\rgroup}\\\\\\\mapsto\sf{\alpha \times \beta =\dfrac{-k}{1} }\\\\\mapsto\sf{\alpha \times \beta =-k }\\\\\mapsto\sf{5 \times (-4) =-k}\\\\\mapsto\sf{\cancel{-} 20=\cancel{-}k}\\\\\mapsto\bf{k=20}$

Thus;

The value of k will be 20 .