Finding roots of equation, with sole parameter.

Question

Find the value of $k$ if product of two roots of equation $$x^4 -37x^3+kx^2 -808x -1984 =0$$ is 62 by using Vieta theorem i can get product of other two root as 32. But what to do after that?

Answer 1

Take care with signs - you need $-32$ not $+32$.

There are various ways of doing this, but one is simply to write: $$(x^2+ax-32)(x^2+bx+62)=x^4+(a+b)x^3+(62-32+ab)x^2+(62a-32b)x-1984$$ and equate coefficients.

$a+b=-37$

$62a-32b=-808$

Solve for $a$ and $b$ and thus find the coefficient of $x^2$.

Answer 2

Let WLOG $$x_1x_2=62$$ then, $$x_3x_4=32$$

Step $1$ $$808=x_1x_2x_3+x_1x_2x_4+x_1x_3x_4+x_2x_3x_4$$ $$=62(x_3+x_4)+32(x_1+x_2)$$

We also know that, $x_1+x_2+x_3+x_4=37$

Thus,

$$x_1+x_2=\frac{62\times37-808}{30}$$

Likewise,

$$x_3+x_4=\frac{808-32\times37}{30}$$

Step $2$ $$k=x_1x_2+x_1x_3+x_1x_4+x_2x_3+x_2x_4+x_3x_4$$

$$=62+32+(x_1+x_2)(x_3+x_4)$$

The answer is not equal to a natural number (if I haven't made a calculation mistake) so I have left it without calculating.