Brush up your fundamentals #3

Algebra Level 1

$\sqrt{2x -1} + \sqrt{3x - 2} = \sqrt{4x -3} + \sqrt{5x -4}$

Find x

Waiting for your method , you will enjoy doing it

The answer is 1.

4 solutions

U Z
Nov 3, 2014

$a = \sqrt{ 2x -1} , b =\sqrt{3x -2} , c = \sqrt{4x -3} , d = \sqrt{ 5x - 4}$

thus

$a + b = c + d$

$a^{2} - b^{2} = 1 - x , c^{2} - d^{2} = 1 - x$

$a^{2} - b^{2} = c^{2} - d^{2}$

thus

$a - b = c -d$

adding

$2a = 2c$

thus x = 1

totally satisfied by your method but one can solve it directly by inspection

on observing carefully, one can see that there is difference of 1 in every expression between the coefficient of x and the constant term.

We have two expressions on both the sides, hence the only solution can be that of x=1.

As simple as that

Rishi Hazra - 6 years, 7 months ago

clever approach, but i am not getting can you please elaborate the last statement @Rishi Hazra

U Z - 6 years, 7 months ago

thanks.................. what I meant to say is that, there are two expressions each on both sides which have a difference of unity between the coefficient of x and the constant term. Thus you can easily go about putting x=1 and obtaining 2=2.

keen observation as you see is indispensable.

Rishi Hazra - 6 years, 7 months ago

Yeah, that helps a lot to make this problem very simple... Very good observation.

Sanjeet Raria - 6 years, 7 months ago

i used observation method to solved this equation :)

Pamela Rose Longhas - 6 years, 6 months ago

Ishan Pradhan
Nov 20, 2014

hit and trial for x=0 it's not defined for x=1 it's true hence ans is 1.

Ranjitha Gowda
Jan 4, 2015

Squaring on both sides & simplifying we get x=1

Asim Das
Dec 19, 2014

putting x=1 we get 2=2. therefore, x=1 is the answer

Brush up your fundamentals #3

The answer is 1.

4 solutions

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