Question

If $\alpha ,\beta$ are the roots of the quadratic equation $a{x}^2+bx+c=0,a\ne 0$. Find the equation with roots $k\alpha +s$ and $k\beta +s$, where k & s are constant and $k\ne 0.$

• A. $a{\left(\frac{x+s}{k}\right)}^2-b\left(\frac{x+s}{k}\right)+c=0$
• B. $a{\left(\frac{x-s}{k}\right)}^2+b\left(\frac{x-s}{k}\right)+c=0$
• C. None of the above
• D. $a{\left(\frac{x+s}{k}\right)}^2+b\left(\frac{x+s}{k}\right)+c=0$

Answer 1

The correct option is B $a{\left(\frac{x-s}{k}\right)}^2+b\left(\frac{x-s}{k}\right)+c=0$

Given: $a{x}^2+bx+c=0,a\ne 0$ with roots $\alpha ,\beta$
To find: Transformed Quadratic equation with roots $k\alpha +s$ and $k\beta +s$

Let, $t=kx+s$

Since, $x=\alpha \Rightarrow t=k\alpha +s$
$x=\beta \Rightarrow t=k\beta +s$

Replace $x\to \frac{t-s}{k}$

$a{\left(\frac{t-s}{k}\right)}^2+b\left(\frac{t-s}{k}\right)+c=0$

In terms of variable x,

$a{\left(\frac{x-s}{k}\right)}^2+b\left(\frac{x-s}{k}\right)+c=0$
This is required equation.