Question

If $\alpha ,\beta$ are the roots of the quadratic equation $a{x}^2+bx+c=0,a\ne 0$. Then the equation with roots $\alpha +k\&\beta +k\forall k\in R$ is:

• A. $a(x+k{)}^2+b(x+k)+c=0$
• B. $a(x-k{)}^2+b(x-k)-c=0$
• C. $a(x-k{)}^2-b(x-k)+c=0$
• D. $a(x-k{)}^2+b(x-k)+c=0$

Answer 1

The correct option is D $a(x-k{)}^2+b(x-k)+c=0$

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

Let, $t=x+k$

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

Replace $x\to t-k$

$a(t-k{)}^2+b(t-k)+c=0$

In terms of variable $x,$ the equation becomes:
$a(x-k{)}^2+b(x-k)+c=0$ which is the required transformed equation.