If x>0 and âˆš5x = âˆš3x+1, what is the value of 4x^2?

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vinni.k: Master | Next Rank: 500 Posts; Posts: 421; Joined: Sun Apr 17, 2011 4:27 amFollowed by: 2 members; Location: India; Thanked: 6 times; GMAT Score:620

by vinni.k » Sun Mar 11, 2018 10:06 am

If x>0 and âˆš5x = âˆš3x+1, what is the value of 4x^2?

A. 1/2
B. 1
C. 4xâˆ’1
D. 10xâˆ’1
E. 16xâˆ’1

OA is E

How to approach this question ?

My approach:-
âˆš5x = âˆš3x+1
Squaring both the sides we get :
5x = 3x + 1 + 2âˆš3x
2x = 1 + 2âˆš3x
Squaring both sides again
4x^2 = 1 + 12x + 4âˆš3x
I am getting stuck here. Any other better approach? Can I plug numbers for x ? The two approaches that are coming into my mind are algebra and number plugging, but both of them seem to be not working at the moment.

Thanks

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Sun Mar 11, 2018 10:17 am

vinni.k wrote:If x>0 and âˆš5x = âˆš3x+1, what is the value of 4x^2?

A. 1/2
B. 1
C. 4xâˆ’1
D. 10xâˆ’1
E. 16xâˆ’1

Since none of the answer choices contains a âˆš, perform algebra that will clear the âˆš symbols:

âˆš(5x) = âˆš(3x) + 1

[âˆš(5x)]Â² = ((âˆš3x) + 1)Â²

5x = 3x + 2âˆš(3x) + 1

2x - 1 = 2âˆš(3x)

(2x - 1)Â² = (2âˆš(3x))Â²

4xÂ² - 4x + 1 = 4(3x)

4xÂ² - 4x + 1 = 12x

4xÂ² = 16x - 1.

The correct answer is E.

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vinni.k: Master | Next Rank: 500 Posts; Posts: 421; Joined: Sun Apr 17, 2011 4:27 amFollowed by: 2 members; Location: India; Thanked: 6 times; GMAT Score:620

by vinni.k » Sun Mar 11, 2018 11:05 am

Thanks Mitch,

I was trying to understand the difference between my approach and your approach. I got stuck with the square roots, but you just got rid of the square roots, after checking the answer choices. I could have got it.

I just want to understand a bit more clearly. In case, if i see a similar kind of question in the test, and the answer choices do contain or don't contain the square roots, then my approach or strategy should be based on what is given in the answer choice.

What if the there were a square root, then my approach could have been right.

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Mon Mar 12, 2018 2:44 am

vinni.k wrote:Thanks Mitch,

I was trying to understand the difference between my approach and your approach. I got stuck with the square roots, but you just got rid of the square roots, after checking the answer choices. I could have got it.

I just want to understand a bit more clearly. In case, if i see a similar kind of question in the test, and the answer choices do contain or don't contain the square roots, then my approach or strategy should be based on what is given in the answer choice.

What if the there were a square root, then my approach could have been right.

We should ALWAYS keep our eyes on the answer choices.
In the posted problem, none of the answers include âˆš, so we should perform algebra that will eliminate this symbol.

vinni.k wrote:2x = 1 + 2âˆš3x
Squaring both sides again
4x^2 = 1 + 12x + 4âˆš3x

Squaring the equation in red will not eliminate the âˆš from the right side.
Before squaring, we should isolate the term with âˆš on the right side, as follows:
2x - 1 = 2âˆš(3x).
Squaring the equation in blue will eliminate the âˆš, leading us to the correct answer.

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deloitte247: Legendary Member; Posts: 2214; Joined: Fri Mar 02, 2018 2:22 pmFollowed by: 5 members

by deloitte247 » Sun Mar 18, 2018 10:12 am

$$Given\ that\ x>0\ \left(i.e\ x\ is\ positive\right)$$
$$\sqrt{5x}=\sqrt{3x+1}$$
$$Square\ both\ sides,$$
$$\left(\sqrt{5x}\right)^2=\left(\sqrt{3x+1}\right)^2$$
$$5x=3x+1$$
$$5x-3x=1$$
$$2x=1,\ hence\ x=\frac{1}{2}\ \left[x\ is\ positive\right]$$
$$Now,\ 4x^2=4\left(\frac{1}{2}\right)^2=4\left(\frac{1}{4}\right)=1\ \left(option\ B\right)$$

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