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 $\alpha -k$ and $\beta -k\forall k\in R$ is

• A. $a(x-k{)}^2+b(x-k)-c=0$
• B. $a(x-k{)}^2+b(x-k)+c=0$
• C. None of the above
• 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, $x=\alpha \Rightarrow t=\alpha -k$
$x=\beta \Rightarrow t=\beta -k$

Replacing $x\to t+k$ in the equation above, we get:

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

Now, In terms of variable $x,$ the equation becomes:

$a(x+k{)}^2+b(x+k)+c=0$ which is the required equation.