Question

I want to ask that if alpha and beta are the zeros of the polynomial PX equal to 2 x square +5x plus k satisfying the relation alpha square + beta square plus alpha beta equals to 21 by 4 then find the value of k for this to possible I want a proper explanation for this question quite confused that why you putalpha + beta square minus 2 alpha beta + alpha beta how it's possible please give me a proper explanation why you take -2 alpha beta

Answer 1

Answer:

k = 2

Step-by-step explanation:

Equation = $2x^2 + 5x + k$

relation with zeroes = $\alpha^2 + \beta^2 + \alpha\beta = \frac{21}{4}$

we will first find out the value of $(\alpha + \beta) \ and \ \alpha\beta$

$\alpha+\beta=\frac{-b}{a} = \frac{-5}{2}\\\alpha\beta = \frac{c}{a} = \frac{k}{2}$

Now we will solve the equation

$\alpha^2 + \beta^2 + \alpha\beta = \frac{21}{4}\\\alpha^2 + \beta^2 + 2\alpha\beta - \alpha\beta = \frac{21}{4}\\(\alpha + \beta)^2 - \alpha\beta = \frac{21}{4}\\Putting \ the \ values \ from \ above\\(\frac{-5}{2})^2 - \frac{k}{2} = \frac{21}{4}\\\frac{25}{4} - \frac{k}{2} = \frac{21}{4}\\\frac{25-2k}{4}=\frac{21}{4}\\25-2k=21\\-2k=21-25\\-2k=-4\\k=2$