Question

The quadratic equation, whose roots are three times the roots of $3a{x}^2+3bx+c=0$, is

• A. $a{x}^2+3bx+3c=0$
• B. $a{x}^2+3bx+c=0$
• C. $9a{x}^2+9bx+c=0$
• D. $a{x}^2+bx+3c=0$

Answer 1

The correct option is A

$a{x}^2+3bx+3c=0$

Find the quadratic equation which has three times the roots of $3a{x}^2+3bx+c=0$,

let us suppose $\alpha ,\beta$ are the roots of the equation $3a{x}^2+3bx+c=0,$

now we need to find the equation whose roots are $3\alpha$ and $3\beta$.

Let us assume $y=3x$,

so,

$$\begin{gathered} y=3x \\ y/3=x \end{gathered}$$

substituting $x$ for $y/3$ in the given equations.

$$\begin{gathered} 3a{(y/3)}^2+3b(y/3)+c=0 \\ \Rightarrow a\left(\right(y^2)/3)+by+c=0 \\ \Rightarrow a{y}^2+3by+3c=0 \\ \Rightarrow a{x}^2+3bx+3c=0 \end{gathered}$$

Hence the quadratic eqaution which has three times the roots of $3a{x}^2+3bx+c=0$, is $a{x}^2+3bx+3c=0$