Question

If $\alpha ,\beta$ are the roots of the equation $a{x}^2+bx+c=0$, then the quadratic equation whose roots are $a\alpha +b$ and $a\beta +b$ is

• A. $x^2-cx+ab=0$
• B. $x^2-bx+ca=0$
• C. $x^2+bx+ca=0$
• D. $x^2+cx+ab=0$

Answer 1

The correct option is B $x^2-bx+ca=0$

Consider the given equation
$a{x}^2+bx+c=0$. Then,
Sum of the roots: $\alpha +\beta =-\frac{b}{a}$

Product of the roots: $\alpha \beta =\frac{c}{a}$
Now,
Sum of the roots of $(a\alpha +b),(a\beta +b)$

$\Rightarrow (a\alpha +b)+(a\beta +b)=(\alpha +\beta )a+2b$

$\Rightarrow (a\alpha +b)+(a\beta +b)=(-\frac{b}{a})a+2b$

$\Rightarrow (a\alpha +b)+(a\beta +b)=-b+2b$

$\Rightarrow (a\alpha +b)+(a\beta +b)=b$

Product of the roots of $(a\alpha +b),(a\beta +b)$

$\Rightarrow (a\alpha +b)(a\beta +b)=a^2\alpha \beta +ab(\alpha +\beta )+b^2$

$\Rightarrow (a\alpha +b)(a\beta +b)$ =$a^2.\frac{c}{a}+ab.\frac{-b}{a}+b^2=ac$

$\therefore$The required quadratic equation is,

$x^2-bx+ac=0$