Q53 / 160 · AP EAMCET 2023 (16 May Shift 2) · Mathematics

If the equation $2 x^3+5 x^2-4 x-12=0$ has a repeated root, then the constant term of the quadratic equation whose roots are the distinct roots of the given equation is

  • Correct answer
    A
    $-6$
  • B
    $-5$
  • C
    $-4$
  • D
    $-2$

Solution

Given equation $2 x^3+5 x^2-4 x-12=0$ $\begin{aligned} & \Rightarrow 2 x^3+4 x^2+x^2+2 x-6 x-12=0 \\ & \Rightarrow\left(2 x^2+x-6\right)(x+2)=0 \\ & \Rightarrow\left(2 x^2+4 x-3 x-6\right)(x+2)=0 \end{aligned}$ $\Rightarrow(2 x-3)(x+2)(x+2)=0 \Rightarrow x=-2,-2, \frac{3}{2}$ so repeated roots $=-2$, Now, quadratic equation $\begin{aligned} & x^2-\left(-2+\frac{3}{2}\right) x+(-2) \times \frac{3}{2}=0 \\ & \Rightarrow 2 x^2+x-6=0 \Rightarrow \text { Constant term }=-6 \end{aligned}$

Hence, the correct answer is option: 'A'.

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