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How many roots does the equation (x^2 + 1)^(1/2) + (x^2 + 2)^(1/2) =

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How many roots does the equation (x^2 + 1)^(1/2) + (x^2 + 2)^(1/2) =  [#permalink]

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New post Updated on: 05 Jan 2020, 04:07
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How many roots does the equation \(\sqrt{x^2 + 1} + \sqrt{x^2 + 2} = 2\) have?


A. 0
B. 1
C. 2
D. 3
E. 4


M12-17

Originally posted by KaranB1 on 05 Jan 2020, 01:43.
Last edited by Bunuel on 05 Jan 2020, 04:07, edited 2 times in total.
Renamed the topic and edited the question.
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Re: How many roots does the equation (x^2 + 1)^(1/2) + (x^2 + 2)^(1/2) =  [#permalink]

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New post 05 Jan 2020, 04:08
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1
KaranB1 wrote:
How many roots does the equation \(\sqrt{x^2 + 1} + \sqrt{x^2 + 2} = 2\) have?


A. 0
B. 1
C. 2
D. 3
E. 4


M12-17


Official Solution:


How many roots does the equation \(\sqrt{x^2 + 1} + \sqrt{x^2 + 2} = 2\) have?


A. 0
B. 1
C. 2
D. 3
E. 4


We know that \(x^2 \ge 0\) so the least value of the left hand side (LHS) of the equation is for \(x=0\):

\(\sqrt{x^2+1} + \sqrt{x^2+2}=1 + \sqrt{2} \approx 2.4\). Now, since even the least possible value of LHS is still greater than RHS (which is 2), then no real \(x\) can satisfy given equation.


Answer: A
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Re: How many roots does the equation (x^2 + 1)^(1/2) + (x^2 + 2)^(1/2) =  [#permalink]

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New post 05 Jan 2020, 04:14
Very nice question.

X^2>=0

So given expression can never be equal to 2, rather it will always be greater than 2.
Ans is 0. A.
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Re: How many roots does the equation (x^2 + 1)^(1/2) + (x^2 + 2)^(1/2) =  [#permalink]

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New post 05 Jan 2020, 04:39
KaranB1 wrote:
How many roots does the equation \(\sqrt{x^2 + 1} + \sqrt{x^2 + 2} = 2\) have?


A. 0
B. 1
C. 2
D. 3
E. 4


M12-17

I think the question should be how many "real" roots does the equation have?
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Re: How many roots does the equation (x^2 + 1)^(1/2) + (x^2 + 2)^(1/2) =  [#permalink]

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New post 05 Jan 2020, 04:40
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Re: How many roots does the equation (x^2 + 1)^(1/2) + (x^2 + 2)^(1/2) =  [#permalink]

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New post 05 Jan 2020, 17:29
Is there a way to do this question using the discriminant?
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Re: How many roots does the equation (x^2 + 1)^(1/2) + (x^2 + 2)^(1/2) =  [#permalink]

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New post 05 Jan 2020, 23:05
letx^2 + 1 be y
sqrt(y) + sqrt(y+1) = 2
sqrt(y) = 2 - sqrt(y+1)
squaring both sides
y= 9/16 i.e. x^2 + 1 = 9/16
x^2= -7/16
square can never be negative .
Hence, no x can satisfy the equation.
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Re: How many roots does the equation (x^2 + 1)^(1/2) + (x^2 + 2)^(1/2) =  [#permalink]

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New post 05 Jan 2020, 23:06
letx^2 + 1 be y
sqrt(y) + sqrt(y+1) = 2
sqrt(y) = 2 - sqrt(y+1)
squaring both sides
y= 9/16 i.e. x^2 + 1 = 9/16
x^2= -7/16
square can never be negative .
Hence, no x can satisfy the equation.
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Re: How many roots does the equation (x^2 + 1)^(1/2) + (x^2 + 2)^(1/2) =   [#permalink] 05 Jan 2020, 23:06
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