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Algebra Level 4

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

Find the number of real solutions to the equation above.

The answer is 1.

1 solution

Archit Tripathi
Sep 17, 2016

Let,

$\sqrt{3x-2}= v$ , $\sqrt{4x+4}= u$ , $\sqrt{5x+1} = p$ and $\sqrt{4x-5} = q$ ,

then we observe,

$u^{2}$ - $v^{2} = x+6 = p^{2} - q^{2} \quad ...(1)$

Since, given that

$u+v = p+q \quad ...(2)$

$u-v = p-q \quad ...(3)$ [from (1)]

Solving $(2)$ and $(3)$ , we get,

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

$\implies x = \boxed{3}$

@Archit Tripathi , I have edited a few of your problems. You don't need to break the LaTex codes with many "\ (" and "\ )". The equations don't appear standards. For example $\sqrt{3x-2} = v$ , notice that the equal sign is bigger and that $v$ is in Roman and italic. $u + v = p + q \quad ...(2)$ , use three dots will do.

Chew-Seong Cheong - 4 years, 9 months ago

Thanks, I will take care next time.

Archit Tripathi - 4 years, 9 months ago

I have edited your solution above. It is easier without keying in so many opening and closing brackets. You should be able to see the LaTex codes using Edit, put your mouse cursor on top of the formulas or use the top right hand pull-down menu and select Toggle LaTex.

Chew-Seong Cheong - 4 years, 9 months ago

Ingenious problem/solution!

James Wilson - 3 years, 5 months ago

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