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How many roots
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04 Nov 2019, 22:43
04 Nov 2019, 22:43
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Question Stats:
50% (01:36) correct
50% (01:48) wrong based on 18 sessions
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Joined: 10 Apr 2015
Posts: 6218
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Re: How many roots
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28 Nov 2019, 10:33
28 Nov 2019, 10:33
1
Asmakan wrote:
How many roots does the equation \(\sqrt{x^2+1}+\sqrt{x^2+2}=2\) have?
Answer:
Show: ::
Question ID: Q02-08
My question: in the explanation, they solve it by substituting zero, why?
My question: in the explanation, they solve it by substituting zero, why?
Answer:
Show: ::
0
Key concept: \(x^2\) will always be greater than or equal to 0.
So, the smallest possible value of \(x^2 + 1\) is \(1\)
This occurs when \(x = 0\)
This means that the smallest possible value of \(\sqrt{x^2+1}\) is 1
Applying the same logic, we can see that the smallest possible value of \(x^2 + 2\) is \(2\)
This means that the smallest possible value of \(\sqrt{x^2+2}\) is \(\sqrt{2}\)
Since \(\sqrt{2} ≈ 1.4\), the smallest possible value of \(\sqrt{x^2+1}+\sqrt{x^2+2}\) ≈ 1 + 1.4 ≈ 2.4
This means it's impossible to have a solution to the equation \(\sqrt{x^2+1}+\sqrt{x^2+2}=2\)
In other words there are no solutions to that equation
Answer: 0
Cheers,
Brent
Re: How many roots
[#permalink]
30 May 2020, 12:05
30 May 2020, 12:05
1
A root of an equation is defined as the x intercept(s) of that equation. SO, this is wherever the graphed line intersects with the x axis, the values for which y=0 and x= something(s).
I've got an unconventional way to think about this problem.
Ignore the square roots on this problem for a second. Just imagine they weren't there. Here's what that graph would look like:
captureA.JPG [ 42.08 KiB | Viewed 2274 times ]
Now, if you add the square roots back into the equation, you're STILL not going to hit that x axis, because if you read the detailed explanation in Greenlighttestprep's answer, there is no real value for which you can manipulate any version of "x^2+1" to hit the x-axis. The "+1" in an equation this simple raises all the values so that there are no intercepts with the x axis.
captureB.JPG [ 43.75 KiB | Viewed 2253 times ]
I've got an unconventional way to think about this problem.
Ignore the square roots on this problem for a second. Just imagine they weren't there. Here's what that graph would look like:
Attachment:
captureA.JPG [ 42.08 KiB | Viewed 2274 times ]
Now, if you add the square roots back into the equation, you're STILL not going to hit that x axis, because if you read the detailed explanation in Greenlighttestprep's answer, there is no real value for which you can manipulate any version of "x^2+1" to hit the x-axis. The "+1" in an equation this simple raises all the values so that there are no intercepts with the x axis.
Attachment:
captureB.JPG [ 43.75 KiB | Viewed 2253 times ]
