How Many Roots Does the Equation \(\sqrt{x^2+1}+\sqrt{x^2+2}=2\) have? GMAT Problem Solving

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

1. 0
2. 1
3. 2
4. 3
5. 4

“How many roots does the equation have \(\sqrt{x^2+1}+\sqrt{x^2+2}=2\) ?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation

Approach Solution 1:

We know that \(x^2\geq0\) so the least value of the left hand side (LHS) of the equation is for x = 0:

\(\sqrt{x^2+1}+\sqrt{x^2+2}=1+\sqrt2\approx2.4\)

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.

Correct Answer: A

Approach Solution 2:

\(\sqrt{x^2+1}+\sqrt{x^2+2}=2\)

\((\sqrt{x^2+1}+\sqrt{x^2+2})^2=2^2\)

\(x^2+1+2(\sqrt{x^2+1})(\sqrt{x^2+2})+x^2+2=4\)

\(2x^2+2(\sqrt{x^4+3x^2+2})=1\)

\(2(x^2+(\sqrt{x^4+3x^2+2}))=1\)

\((x^2+(\sqrt{x^4+3x^2+2}))=\frac{1}{2}\)

\((\sqrt{x^4+3x^2+2})=\frac{1}{2}-x^2\)

\(x^4+3x^2+2=(\frac{1}{2}-x^2)^2\)

\(x^4+3x^2+2=\frac{1}{4}-x^2+x^4\)

\(4x^2+\frac{7}{4}=0\Rightarrow16x^2+7=0\)

This is a quadratic equation of the form \(ax+bx+c=0\) where a = 16, b = 0, c = 7

The discriminant of a quadratic equation \(D=b^2-4ac\)

When D > 0, there are exactly two real solutions
When D = 0, there is exactly one real solution
When D < 0, there are no real solutions

In this case \(D=0^2-4(16)(7)<0\)

So there are no real solutions

Correct Answer: A

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