Question

The equation whose roots are the squares of the roots of the equation $2x^2+3x+1=0$, is:

• A. $4x^2+5x+1=0$
• B. $4x^2-x+1=0$
• C. $4x^2-5x-1=0$
• D. $4x^2-5x+1=0$

Answer 1

The correct option is D

$4x^2-5x+1=0$

Explanation for the correct option:

Step 1: Find the roots of the quadratic equation.

A quadratic equation $2x^2+3x+1=0$ is given.

Use quadratic formula to find the roots of the given quadratic equation.

$$\begin{gathered} x=\frac{-3\pm \sqrt{3^2-4·2·1}}{2·2} \\ \Rightarrow x=\frac{-3\pm \sqrt{9-8}}{4} \\ \Rightarrow x=\frac{-3\pm \sqrt{1}}{4} \\ \Rightarrow x=\frac{-3\pm 1}{4} \\ \Rightarrow x=\left\{-1,\frac{-1}{2}\right\} \end{gathered}$$

Therefore, the roots of the given quadratic equation are $-1,\frac{-1}{2}$.

Step 2: Find the required quadratic equation.

Since, the roots of the required quadratic equation are equal to the square of given quadratic equation.

That is, the roots of the required quadratic equation are $1,\frac{1}{4}$.

So, the required quadratic equation can be given as:

$$\begin{gathered} (x-1)\left(x-\frac{1}{4}\right)=0 \\ \Rightarrow x^2-\frac{1}{4}x-x+\frac{1}{4}=0 \\ \Rightarrow 4x^2-x-4x+1=0 \\ \Rightarrow 4x^2-5x+1=0 \end{gathered}$$

Therefore, the quadratic equation whose roots are the squares of the roots of the equation $2x^2+3x+1=0$, is $4x^2-5x+1=0$.

Hence, option $D$ is the correct.