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Algebra Level 3

$\big(\sqrt{x+1}+1\big)\big(\sqrt{x+16}+4\big)-\big(\sqrt{x+4}+2\big)\big(\sqrt{x+9}+3\big)=0$

How many real value(s) of $x$ satisfy the above equation?

None of the given choices. 1 2 4 0

2 solutions

Vaibhav Prasad
May 31, 2015

$(\sqrt{x+1}+1)(\sqrt{x+16}+4)-(\sqrt{x+4}+2)(\sqrt{x+9}+3)=0$

$\rightarrow (\sqrt{x+1}+1)(\sqrt{x+16}+4)=(\sqrt{x+4}+2)(\sqrt{x+9}+3)$

So, the values of $x$ that will satisfy the above equation will the points that coincide when the graphs of $LHS$ and $RHS$ are drawn.

The graphs are found to be parallel to each other

Hence there do not exist any solutions for $x$

Moderator note:

You have only shown that there is no solution in a finite interval, can you show that there is no solution throughout the whole domain?

And, can you solve this without graphing?

Is there any other way to solve this .....?

Samarth Agarwal - 6 years ago

There sure is, but it would involve tedious algebra and the application of many formulas...why do that when there is such an easy way to solve it! :)

Aneesh S. - 6 years ago

Yes, just look at the RHS and LHS. Two irrational quantities can be made equal only by making the rational and irrational parts equal individually, like in the case of complex numbers. Also, from the expression you can say the irrational parts will never be equal. Hence, no solution.

Yugesh Kothari - 5 years, 11 months ago

guys give me credit plzzz i was first to solve here.

-1- ( (x+1)^(1/2) + 1 )( (x+16)^(1/2) + 4 ) - ( (x+4)^(1/2) + 2 )( (x+9)^(1/2) + 3 ) = 0

A= ( (x + (a^2) )^(1/2) +a ) = x/[( (x + (a^2) )^(1/2) - a )] taking x common and manipulating eqn -1- little bit

-2- ( (x+1)^(1/2) - 1 )( (x+16)^(1/2) - 4 ) - ( (x+4)^(1/2) - 2 )( (x+9)^(1/2) - 3 ) = 0

-3- opening brackets for eqn 1 and 2 and adding both of them we get:- [(x+1)(x+16)] + 4 -[(x+4)(x+9)]-6=0

-4- after solving we may get x==0 rejecting that solution we have no other fundamental soln..

btw sry i pressed the wrong answer by mistake

Nikhil Jay - 4 years, 1 month ago

Man I didn't understand it can you plz be a bit elaborate

Sohail Sarkar - 2 years, 10 months ago

Niranjan Khanderia
May 31, 2015

The graph of the
$y=(\sqrt{x+1}+1)(\sqrt{x+16}+4)-(\sqrt{x+4}+2)(\sqrt{x+9}+3)$ 0n the calculator shows that y=0 is the asymptote of the curve. So y=0 is never reached. Implies no solution. the window was set to x= 12 * 7 and y=-.001.

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