The real roots of the equation $\sqrt{x}+\sqrt{x-\sqrt{1-x}}=1$ is/are
- $0$
- $\dfrac{16}{25}$
- $2$
- $3$
Reveal answer
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B
Correct answer
Explanation
Let sqrt(x) = u. Then u + sqrt(u^2 - sqrt(1-u^2)) = 1. Testing options, for 16/25, sqrt(16/25) = 4/5. sqrt(1-16/25) = sqrt(9/25) = 3/5. Equation becomes 4/5 + sqrt(16/25 - 3/5) = 4/5 + sqrt(1/25) = 4/5 + 1/5 = 1. Correct.
AI explanation
Square both sides to get x + x - sqrt(1-x) + 2sqrt(x^2 - x*sqrt(1-x)) = 1. Rearranging the terms under the square root and squaring again systematically yields a solvable polynomial equation for x. Testing the provided positive value by substituting x = 16/25 back into the original equation results in sqrt(16/25) + sqrt(16/25 - sqrt(9/25)) = 4/5 + sqrt(16/25 - 3/5) = 4/5 + sqrt(1/25) = 4/5 + 1/5 = 1, proving x = 16/25 is the correct root.