Question

The number of positive integral solutions $x^2+9<(x+3{)}^2<8x+25$, is

• A. $2$
• B. $3$
• C. $4$
• D. $5$

Answer 1

Answer: (D)

$5$

$x^2+9<(x+3{)}^2<8x+25\Rightarrow x^2+9<(x+3{)}^2<9\Rightarrow x^2+9<x^2+9+6x\Rightarrow 6x>0\Rightarrow x>0$

Also $(x+3{)}^2<8x+25$

$\Rightarrow x^2-2x-16<0\Rightarrow 1-\sqrt{17}<x<1+\sqrt{17}$

But $x>0$

$\Rightarrow 0<x<1+\sqrt{17}\Rightarrow x\in (0,1+\sqrt{17})$

So the integer values in our domain are $1,2,3,4,5$

So, option D is correct.