\-x+sqrt1−x 2 \-=sqrt2(2x 2 −1).
Giving extra points for the best answer

x = -√½ or √⁹/₁₀

Step-by-step explanation:

x + √(1 − x²) = √(2(2x² − 1))

Square both sides:

x² + 2x√(1 − x²) + 1 − x² = 2(2x² − 1)

2x√(1 − x²) + 1 = 4x² − 2

2x√(1 − x²) = 4x² − 3

Square both sides again:

4x² (1 − x²) = 16x⁴ − 24x² + 9

4x² − 4x⁴ = 16x⁴ − 24x² + 9

0 = 20x⁴ − 28x² + 9

Solve with quadratic formula:

x² = [ 28 ± √(28² − 4(20)(9)) ] / 2(20)

x² = (28 ± 8) / 40

x² = ½, ⁹/₁₀

x = ±√½, ±√⁹/₁₀

Check for extraneous solutions.

If x = -√½:

-√½ + √(1 − ½) = √(2(2(½) − 1))

-√½ + √½ = √(2(1 − 1))

0 = 0

If x = √½:

√½ + √(1 − ½) = √(2(2(½) − 1))

√½ + √½ = √(2(1 − 1))

√2 = 0

If x = -√⁹/₁₀:

-√⁹/₁₀ + √(1 − ⁹/₁₀) = √(2(2(⁹/₁₀) − 1))

-√⁹/₁₀ + √¹/₁₀ = √(2(⁹/₅ − 1))

-√⅖ = √⁸/₅

If x = √⁹/₁₀:

√⁹/₁₀ + √(1 − ⁹/₁₀) = √(2(2(⁹/₁₀) − 1))

√⁹/₁₀ + √¹/₁₀ = √(2(⁹/₅ − 1))

√⁸/₅ = √⁸/₅

898.1666667

step-by-step explanation:

5389/6 = 898.1666667

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