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If $x+1=2014^{2}+2015^{2}$ , what is $\sqrt{2x+1}$ ?

The answer is 4029.

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Oh... I think you might have a better solution than I do... I didn't factor...

Finn Hulse
- 7 years, 4 months ago

Dear Victor

Please state your source for this question and all questions you post

Wei Jie Tan
- 7 years, 4 months ago

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Sure, 'D D'

Victor Loh
- 7 years, 4 months ago

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You know who I am?

Wei Jie Tan
- 7 years, 4 months ago

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Of course. Do you mind me stating your name?

Victor Loh
- 7 years, 4 months ago

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@Victor Loh – You still did not state the source

Wei Jie Tan
- 7 years, 4 months ago

I know you hate this question XD

Victor Loh
- 7 years, 4 months ago

1 Helpful
0 Interesting
0 Brilliant
0 Confused

**
TO GENERALIZE THE PROBLEM
**

**
Consider 2 numbers
$a, b$
where
$b=a+1$
**

$x+1=a^{2}+b^{2}$

**
substitute
$b=a+1$
**

$x+1=a^{2}+(a+1)^{2}$

$x+1=a^{2}+a^{2}+1+2a$

$x=2a^{2}+2a$

**
Now substitute the value of x in given equation
**

$\sqrt{2x+1}$

$\sqrt{2(2a^{2}+2a)+1}$

$\sqrt{4a^{2}+4a+1}$

**
If you observe, the given equation can be factorised
**

$\sqrt{(2a+1)^{2}}$

$2a+1$

$a+a+1$

ie $a+b$

**
So the value of
$\sqrt{2x+1} = a+b$
**

Ans $\boxed{4029}$

Anirudha Nayak
- 7 years, 4 months ago

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This is a lot easier than mine, almost no calculations.

May Lai
- 7 years, 4 months ago

I'm just kinda confused that how you broke it into two parts, could you explain it to me?

May Lai
- 7 years, 4 months ago

×

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First we move the +1 to the right side and we get x=2014^2+2015^2-1 Factoring 2015^2-1 we get (2015+1)(2015-1) which equals 2016x2014 For x=2014^2+2014x2016 we take 2014 out. Thus we get x=2014(2014+2016)=2014x4030 2x=2x2014x4030=4028x4030=(4029-1)(4029+1)=4029^2-1 2x+1=4029^2 Therefore the square root of 2x+1 equals to 4029

I'll never use iPad to type those numbers and signs again:-(