Question

if alpha, beta are the zeros of x square - 2 X + 3 k such that alpha plus beta equals to Alpha Beta then K equals to​

Answer 1

The value of k is  $\dfrac{2}{3}$ .

Step-by-step explanation:

We are given that alpha, beta are the zeros of $x^{2} -2x+3k$ such that alpha plus beta equal to Alpha Beta.

We have to find the value of k from the equation.

As we know that the quadratic equation having zeroes alpha($\alpha$) and beta($\beta$) is given by;

$ax^{2}+bx+c$  having zeroes $\alpha$ and $\beta$ where;

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

Now, the quadratic equation given in the question is;

$x^{2} -2x+3k$

Here, a = 1, b = -2 and c = 3k.

So, the sum of the zeroes ($\alpha + \beta$)  =  $\frac{-b }{a}$

                        $\alpha + \beta = \frac{-(-2)}{1}$

and, the product of zeroes ($\alpha \times \beta$)  =  $\frac{c}{a}$

                        $\alpha \times \beta = \frac{3k}{1}$

Since it is given that $\alpha + \beta = \alpha \times \beta$, this means;

                          $\frac{2}{1}=\frac{3k}{1}$

                          $k=\frac{2}{3}$

Hence, the value of k is  $\frac{2}{3}$.

Answer 2

Given:

$x^2-2x+3k\\\\ \alpha\ \ _{and} \ \ \beta \ \ \ are \ zeros$

To find:

$\alpha +\beta =\alpha \beta$ then

k=?

Solution:

Compare the given value with the standard equation:

Equation:

$x^2-2x+3k$

Standard Equation:

$ax^2+bx+c=0$

compare the value:

$a=1\\b=-2\\c=3k\\\\ \alpha+\beta = \frac{-b}{a}= \frac{-(-2)}{1}= 2\\\\\alpha \beta =\frac{c}{a}= \frac{3k}{1}= 3k$

$\to \alpha +\beta = 2\\\\\to\alpha \beta = 3k$

when:

$\to \bold{ \alpha +\beta =\alpha \beta}\\\\\to 2=3k\\\\\to \boxed{k= \frac{2}{3}}$