Solution: Square both sides: $ x^2 + 2 + \frac{1}{x^2} = 25 \Rightarrow x^2 + \frac{1}{x^2} = 23 $. Multiply by 2: $ 2x^2 + \frac{2}{x^2} = 46 $. Final answer: $ \boxed{46} $.Question: In a sustainable desert agriculture project, the combined water efficiency of two systems is modeled by $ a + b = 10 $ and $ a^2 + b^2 = 58 $. Determine the total water efficiency cubed, $ a^3 + b^3 $.

["Title: Calculating Total Water Efficiency Cubed in Desert Agriculture Systems Using Algebra", "In sustainable desert agriculture, optimizing water efficiency between two integrated systems is crucial. Given the constraints $ a + b = 10 $ and $ a^2 + b^2 = 58 $, scientists aim to compute $ a^3 + b^3 $—a measure of total efficient water usage across both systems.", "This problem involves algebraic identities that yield powerful insights with minimal computation—ideal for resource-conscious research environments.", "We begin by using the identity for the sum of cubes:", "$$\na^3 + b^3 = (a + b)^3 - 3ab(a + b)\n$$", "We already know $ a + b = 10 $, so compute $ (a + b)^3 $:", "$$\n(10)^3 = 1000\n$$", "To apply the identity, we need $ ab $. This can be found using the given $ a^2 + b^2 = 58 $ and the square of the sum:", "$$\n(a + b)^2 = a^2 + 2ab + b^2\n$$", "Substitute known values:", "$$\n10^2 = 58 + 2ab \Rightarrow 100 = 58 + 2ab \Rightarrow 2ab = 42 \Rightarrow ab = 21\n$$", "Now substitute into the sum of cubes formula:", "$$\na^3 + b^3 = 1000 - 3(21)(10) = 1000 - 630 = 370\n$$", "Wait—this contradicts the expected insight. Let’s reevaluate using an alternative identity that directly connects $ a^2 + b^2 $, $ a + b $, and $ a^3 + b^3 $ via intermediate steps.", "An efficient shortcut uses:", "$$\na^3 + b^3 = (a + b)^3 - 3ab(a + b)\n$$", "But to avoid double-counting identities, let’s instead derive $ a^3 + b^3 $ through known algebraic manipulation.", "Alternatively, recall:", "$$\na^3 + b^3 = (a + b)\left(a^2 - ab + b^2\right)\n$$", "We already have $ a + b = 10 $, $ a^2 + b^2 = 58 $, and $ ab = 21 $. Then:", "$$\na^2 - ab + b^2 = (a^2 + b^2) - ab = 58 - 21 = 37\n$$", "Thus:", "$$\na^3 + b^3 = 10 \ imes 37 = 370\n$$", "This confirms:", "$$\n\boxed{370}\n$$", "This precise algebraic derivation enables researchers to model total cubic water efficiency with accuracy and minimal environmental data burden—key in arid-zone sustainability.", "In sustainable desert farming, leveraging compact equations like $ a + b = 10 $, $ a^2 + b^2 = 58 $, and deriving $ a^3 + b^3 = 370 $, allows engineers to validate system synergy efficiently and scale resilient irrigation strategies with mathematical confidence."]