MathCounts 2026 Chapter Sprint Round (Problem 30)If the real number xxx satisfies the equation x2+x+x2−x=2\sqrt{x^2+\sqrt{x}}+\sqrt{x^2-\sqrt{x}}=2x2+x+x2−x=2, what is the value of 8x8x8x? Express your answer in simplest radical form.Related TopicsCoreToolkit 69 — Two Approaches for Square RootsMajorToolkit 21 — Quadratic FormulaMinorToolkit 10 — Difference of squaresToolkit 6 — Square of a sumHints (4)Hint 1x2+x+x2−x=2\sqrt{x^2+\sqrt{x}}+\sqrt{x^2-\sqrt{x}}=2x2+x+x2−x=2Square both sides:x2+x+x2−x+2x4−x=4x^2+\sqrt{x}+x^2-\sqrt{x}+2\sqrt{x^4-x}=4x2+x+x2−x+2x4−x=42x2+2x4−x=42x^2+2\sqrt{x^4-x}=42x2+2x4−x=4x2+x4−x=2x^2+\sqrt{x^4-x}=2x2+x4−x=2Hint 2x4−x=2−x2\sqrt{x^4-x}=2-x^2x4−x=2−x2Square both sides:x4−x=(2−x2)2=4+x4−4x2x^4-x=(2-x^2)^2=4+x^4-4x^2x4−x=(2−x2)2=4+x4−4x24x2−x−4=04x^2-x-4=04x2−x−4=0Hint 3By Toolkit 21 — Quadratic Formula:x=−(−1)±(−1)2−4(4)(−4)2(4)=1±658x=\frac{-(-1)\pm\sqrt{(-1)^2-4(4)(-4)}}{2(4)}=\frac{1\pm\sqrt{65}}{8}x=2(4)−(−1)±(−1)2−4(4)(−4)=81±65Hint 4Since there is x\sqrt{x}x,x≥0⇒x=1+658x\ge0\Rightarrow x=\frac{1+\sqrt{65}}{8}x≥0⇒x=81+65⇒8x=1+65\Rightarrow 8x=1+\sqrt{65}⇒8x=1+65Final Answer1+651+\sqrt{65}1+65Related Problems (2)MATHCOUNTS State Sprint 2025 (Problem 26)AMC 12A Fall 2021 (Problem 19)