Question

What are the zeros of the polynomial x2-6x+k. What is the value of k of 3alpha+2beta=20.

Answer 1

AnsWer :

k = -16.

Given :

• We have expression x² -6x + k.
• 3 alpha + 2beta = 20.

To Find :

Value of k.

Concepts Required :

$\tt \blacksquare \: Sum \: of \: Zeros. \\ \tt \alpha + \beta = \frac{ - b}{a}$

$\tt \blacksquare \: Product \: of \: Zeros. \\ \tt \alpha . \beta = \frac{ c}{a}$

Solution :

We have,

$\tt \: \: 3 \alpha + 2 \beta = 20. - - - (1)$

Sum of zeros,

$\tt \: \: \alpha + \beta = \frac{ - b}{a} = 6 - - - (2)$

And,

Product of zeros,

$\tt \: \: \alpha . \beta = \frac{c}{a} = k - - - (3)$

$\rule{200}3$

Now,

Taking equation (2)

$\tt \implies \alpha + \beta = 6$

Multiply by 2, We get.

$\tt \implies2 \alpha +2 \beta = 12. - - -( 4)$

By equation 1 Subtracted From equation 4, We get.

$\begin{tabular}{1-1} 3 \alpha + 2 \beta= 20 & \\ 2 \alpha +2 \beta = 12 & \\ \cline{1-1} \alpha = 8 & \\ \cline{1-1} \end{tabular}$

$\tt \implies \alpha = 8.$

Putting the value of alpha in equation 2.

$\tt \implies\alpha + \beta = 6.$

$\tt \implies8 + \beta = 6.$

$\tt \implies \beta = - 2.$

$\rule{200}3$

Now, According to Question.

Putting the value of alpha and beta in equation 3, We get the value of k.

$\tt \implies \alpha \beta = k$

$\tt \implies8 \times - 2 = k.$

$\tt \implies k = - 16.$

Therefore,the value of k is -16.

Answer 2

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