If $a<b<c<d$ then predict the nature of roots of the equation $3(x-a)(x-c)+5(x-b)(x-d)=0$

Question

I’ve tried to use discriminant method, but it looked quite complex.

The answer to this question has to be expressed as real/imaginary and distinct/equal roots.

Answer 1

Hint. Let $f(x)=3(x-a)(x-c)+5(x-b)(x-d)$ (which is polynomial of second degree). Then $$f(a)=5(a-b)(a-d)>0,\quad f(b)=3(b-a)(b-c)<0,\\ f(c)=5(c-b)(c-d)<0, \quad f(d)=3(d-a)(d-c)>0.$$

Answer 2

HINT: the discriminat is given by $$(-3a-5b-3c-5d)^2-32(3ac+5bd)$$

Answer 3

Hint: this is clearly a second degree polynomial with leading term $8x^2$, so the values will be large and positive if $x$ is either large and positive or large and negative.

It will have therefore real roots if there is a value of $x$ for which the polynomial is negative (or zero). There are some obvious values to try.