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If x>0 and √5x = √3x+1, what is the value of 4x^2?

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by vinni.k » Sun Mar 11, 2018 10:06 am
If x>0 and √5x = √3x+1, what is the value of 4x^2?

A. 1/2
B. 1
C. 4x−1
D. 10x−1
E. 16x−1

OA is E

How to approach this question ?

My approach:-
√5x = √3x+1
Squaring both the sides we get :
5x = 3x + 1 + 2√3x
2x = 1 + 2√3x
Squaring both sides again
4x^2 = 1 + 12x + 4√3x
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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by GMATGuruNY » Sun Mar 11, 2018 10:17 am
vinni.k wrote:If x>0 and √5x = √3x+1, what is the value of 4x^2?

A. 1/2
B. 1
C. 4x−1
D. 10x−1
E. 16x−1
Since none of the answer choices contains a √, perform algebra that will clear the √ symbols:

√(5x) = √(3x) + 1

[√(5x)]² = ((√3x) + 1)²

5x = 3x + 2√(3x) + 1

2x - 1 = 2√(3x)

(2x - 1)² = (2√(3x))²

4x² - 4x + 1 = 4(3x)

4x² - 4x + 1 = 12x

4x² = 16x - 1.

The correct answer is E.
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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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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 √, so we should perform algebra that will eliminate this symbol.
vinni.k wrote:2x = 1 + 2√3x
Squaring both sides again
4x^2 = 1 + 12x + 4√3x
Squaring the equation in red will not eliminate the √ from the right side.
Before squaring, we should isolate the term with √ on the right side, as follows:
2x - 1 = 2√(3x).
Squaring the equation in blue will eliminate the √, leading us to the correct answer.
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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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