If the product of 2 roots of eqution is -31 then find the value of k
Solution
Product of 4 roots = -2015
product of 2 roots = -31
so product of other 2 roots = -2015/(-31) = 65
so 2 quadratic factors are $x^2+ax-31$ and $x^2+bx +65$ where a and b are to be determined
so $x^4-18x^3+kx^2+174x - 2015=(x^2+ax-31)(x^2+bx +65)$
or $ x^4 + (a+b)x^3 +(65-31+ab)x^2 + (65a - 31b)x - 2015=0$
comparing coefficients we get
$a+b= -18$, $(65a-31b= 174$,$k= 34 + ab$
we can solve 1st 2 to get $a= -4, b= -14$ so putting in 3rd we get $k= 34 + ab= 90$
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