Question

For the equation
$2x^4-3x^3-x^2-3x+2=0$,
Which among the following statement(s) is/are correct?

• A. Sum of the real roots = $\frac{5}{2}$
• B. Product of real roots = $1$.
• C. All roots of the equation are real
• D. Exactly two roots of the equation are real

Answer 1

Answer: (A), (B), (D)

Exactly two roots of the equation are real

Given equation
$2x^4-3x^3-x^2-3x+2=0$
Dividing the equation by $x^2$,
$2(x^2+\frac{1}{x^2})-3(x+\frac{1}{x})-1=0$ $\Rightarrow 2({(x^2+\frac{1}{x^2})}^2-2)-3(x+\frac{1}{x})-1=0$
Assuming
$x+\frac{1}{x}=t\Rightarrow t\in R-(-2,2)$

Then,
$2(t^2-2)-3t-1=0$
$\Rightarrow 2t^2-3t-5=0\Rightarrow (t+1)(2t-5)=0\Rightarrow t=-1,\frac{5}{2}\therefore t=\frac{5}{2}$

Now,
$x+\frac{1}{x}=\frac{5}{2}\Rightarrow 2x^2-5x+2=0\Rightarrow (2x-1)(x-2)=0\Rightarrow x=2,\frac{1}{2}$