Question

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

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

Answer 1

The correct option is C

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

Explanation for the correct option:

Find the quadratic equation:

Let $\alpha ,\beta$ be the roots of the equation $3a{x}^2+3bx+c=0.$

We have to find the equation whose roots are $3\alpha$ and $3\beta$,

Substitute $x=\frac{y}{3}$ into the given equation, we get

$$\begin{gathered} \Rightarrow 3a{\left(\frac{y}{3}\right)}^2+3b\left(\frac{y}{3}\right)+c=0 \\ \Rightarrow 3a\frac{y^2}{9}+yb+c=0 \\ \Rightarrow \frac{a{y}^2}{3}+yb+c=0 \\ \Rightarrow a{y}^2+3yb+3c=0...\left(1\right) \end{gathered}$$

We can write equation $\left(1\right)$ in term of $x$. then

$\Rightarrow a{x}^2+3xb+3c=0$

Hence, the correct option is C.