Asmakan wrote:
How many roots does the equation \(\sqrt{x^2+1}+\sqrt{x^2+2}=2\) have?
Question ID: Q02-08
My question: in the explanation, they solve it by substituting zero, why?
Answer:
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