Question

If $\alpha and\beta$ are the roots of the equation $x^2-2x+3=0$ Find the equation whose roots are
$\alpha +2,\beta +2$

• A. $x^2-3x+11=0$
• B. $x^2+6x+11=0$
• C. $x^2-6x+11=0$
• D. $x^2+3x+11=0$

Answer 1

The correct option is C $x^2-6x+11=0$

$\Rightarrow$ The given quadratic equation is $x^2-2x+3=0$, comparing it with $a{x}^2+bx+c=0$

$\Rightarrow$ We get, $a=1,b=-2,c-3$

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

$\therefore$ $\alpha +\beta =\frac{-(-2)}{1}=2$ ----- ( 1 )

$\Rightarrow$ $\alpha \beta =\frac{c}{a}=\frac{3}{1}=3$ ---- ( 2 )

$\Rightarrow$ Now, $(\alpha +2)+(\beta +2)=\alpha +\beta +4$

$\therefore$ $(\alpha +2)+(\beta +2)=2+4=6$ [ From ( 1 ) ] ---- ( 3 )

$\Rightarrow$ $(\alpha +2)(\beta +2)=\alpha \beta +2(\alpha +\beta )+4$

$\therefore$ $(\alpha +2)(\beta +2)=3+2\left(2\right)+4=11$ [ From (1 ) and ( 2 ) ] --- ( 4 )

$\Rightarrow$ The new equation, $x^2-\left[\right(\alpha +2)+(\beta +2\left)\right]+\left[\right(\alpha +2\left)\right(\beta +2\left)\right]=0$

From ( 3 ) and ( 4 ),

$\therefore$ $x^2-6x+11=0$