MathLabs

Problem 1

For which real values of pp does the equation x2−p+2x2−1=x\sqrt{x^2-p}+2\sqrt{x^2-1}=x have real roots? What are the roots?
Step 5 of 5: Verify the root
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In plain words

The domain and sign checks select one positive root and reject all extraneous squared roots.

0≤p≤43:x=4−p16−8p0\le p\le\frac43:\qquad x=\frac{4-p}{\sqrt{16-8p}}
Detailed analysis

For this interval the denominator is positive and xx must be the positive square root, so x=(4−p)/16−8px=(4-p)/\sqrt{16-8p}. Substitution into the original equation (or the equivalent absolute-value check) confirms it for every pp in the interval; outside it no solution exists.