Question

Solve the equation $\sqrt{2x-6}+\sqrt{x+4}=5$

• A. 2
• B. 4
• C. 5
• D. 8

Answer 1

Answer: (C)

$\sqrt{2x-6}+\sqrt{x+4}=5$ Here $2x-6>0$ means $x>3$

and also $x+4>0$ which means $x>-4$ $x$ need to satisfy these conditions also

$\Rightarrow$ $\sqrt{2x-6}=-\sqrt{x+4}+5$ ----- ( 1 )

$\Rightarrow$ $(\sqrt{2x-6}{)}^2=(-\sqrt{x+4}+5{)}^2$ [ Squaring both sides ]

$\Rightarrow$ $2x-6=x+4+25-10\sqrt{x+4}$

$\Rightarrow$ $10\sqrt{x+4}=-2x+6+x+4+25$

$\Rightarrow$ $10\sqrt{x+4}=-x+35$

$\Rightarrow$ $(10\sqrt{x+4}{)}^2=(-x+35{)}^2$ [ Squaring both sides ]

$\Rightarrow$ $100x+400=x^2-70x+1225$

$\Rightarrow$ $x^2-170x+825=0$

$\Rightarrow$ $x^2-165x-5x+825=0$

$\Rightarrow$ $x(x-165)-5(x-165)=0$

$\Rightarrow$ $(x-165)(x-5)=0$

$\therefore$ $x=165$ or $x=5$

Now substitute $x=165$ in equation (1 ),

$\Rightarrow$ $\sqrt{(2\times 165)-6}=-\sqrt{\left(165\right)+4}+5$

$\Rightarrow$ $\sqrt{324}=-8$

$\Rightarrow$ $18\ne -8$

Now substituting $x=5$ in equation ( 1 ),

$\Rightarrow$ $\sqrt{(2\times 5)-6}=\sqrt{5+4}+5$

$\Rightarrow$ $\sqrt{4}=2$

$\therefore$ $2=2$

$\therefore$ Solution is $x=5$