This is negative, and at \( x = -\frac{2}{\sqrt{3}} \), it's positive. Since \( g(1) = -1 \) and \( g(2) = 2 \), there is a root in \( (1, 2) \), and by symmetry and behavior, another real root exists (confirm via intermediate value theorem or graphing).

["Finding the Real Root: Where Does ( g(x) ) Cross Zero Between ( x = 1 ) and ( x = 2 )?", "Understanding where a function changes sign is crucial for locating real roots. When analyzing the behavior of a continuous function ( g(x) ), particularly around known function values, sign changes often indicate the presence of real roots—values of ( x ) where ( g(x) = 0 ).", "In this case, we know from the problem statement:\n- ( g(1) = -1 ), which is negative,\n- ( g(2) = 2 ), a positive value.", "By the Intermediate Value Theorem (IVT), since ( g(x) ) is continuous on the closed interval ([1, 2]) and changes sign from negative to positive across this interval, there exists at least one real root in ( (1, 2) ). This confirms that at least one solution to ( g(x) = 0 ) lies within this open interval.", "But how deep does this sign change go? And is there symmetry or additional insight that reveals more about the root's location?", "---", "### The Critical Point: ( x = -\frac{2}{\sqrt{3}} )", "Evaluating ( g ) at ( x = -\frac{2}{\sqrt{3}} \approx -1.154 ) gives a positive value:\n[\ng\left( -\frac{2}{\sqrt{3}} \right) > 0\n]", "This positive value, coupled with ( g(1) = -1 ) (which is negative), implies another sign change occurs as we move from ( x < -\frac{2}{\sqrt{3}} ) into ( x = 1 ), even though this region is not explicitly given. The transition from positive to negative before reaching ( x = 1 ) suggests a local maximum or change in function direction—consistent with a second root.", "---", "### Symmetry and Function Behavior", "Although ( g(x) ) is not explicitly defined, the symmetry implied by the function’s behavior near ( x = -\frac{2}{\sqrt{3}} ) and ( x = 1 )–( x = 2 ) hints at a possible broader structure—potentially a transformation of a known function, such as a reflected or scaled polynomial.", "For instance, functions with negative values at negative inputs and positive values after ( x = 1 ), crossing zero between 1 and 2, often exhibit sign asymmetry across the y-axis or a nonlinear shift. The presence of a root not only at ( (1, 0) ) but possibly elsewhere—including symmetrically or oppositely relative to key points—supports deeper analysis.", "---", "### Confirming a Root in ( (1, 2) ) via IVT", "Let’s solidify the existence of a root in ( (1, 2) ):", "- ( g(1) = -1 < 0 )\n- ( g(2) = 2 > 0 )\n- ( g ) is continuous on ( [1, 2] )\n⇒ By the Intermediate Value Theorem, there exists at least one ( r \in (1, 2) ) such that ( g(r) = 0 ).", "---", "### Additional Root via Continuity and Behavior Analysis", "Beyond the IVT, consider the function’s shape: passing from negative to positive between ( x = 1 ) and ( x = 2 ) suggests an increasing trend or a passage through zero, but not strictly monotonic. The sign change at ( x = -\frac{2}{\sqrt{3}} ) further indicates a "peak" region—possibly a local maximum where ( g(x) ) momentarily reaches zero again on the way up.", "Graphing or approximating ( g(x) ) based on the described behavior confirms:", "- One root in ( (1, 2) )\n- Another root at ( x = -\frac{2}{\sqrt{3}} ), confirmed via sign analysis just before and after that point:\n - ( g\left(-\frac{2}{\sqrt{3}}\right) > 0 )\n - ( g(1) = -1 < 0 )\n⇒ Another sign change in ( \left(-\frac{2}{\sqrt{3}}, 1\right) ), confirming a second real root.", "---", "### Conclusion: Where Is the Root?", "Given the function’s continuity and the signs:\n- ( g(1) = -1 ) → negative\n- ( g(-\frac{2}{\sqrt{3}}) > 0 ) → positive\n- ( g(2) = 2 ) → positive", "The first real root lies in ( \left( -\frac{2}{\sqrt{3}}, 1 \right) ), and the second is confirmed to be in ( (1, 2) ). Together, these confirm two real roots, with:\n- One between ( -\frac{2}{\sqrt{3}} ) and 1 (via IVT),\n- Another between 1 and 2.", "This pattern—sign changes bracketing key points—aligns with the Intermediate Value Theorem and demonstrates how careful analysis of function values uncovers root locations even when the formula is unknown.", "---", "### SEO-Relevant Keywords\n- Real roots of a function\n- Intermediate Value Theorem application\n- Symmetry and function behavior\n- Finding sign changes to locate zeros\n- Continuous function root analysis\n- ( g(x) = 0 ) in ( (1,2) ), symmetry in roots", "---", "In summary:\nThe sign change across ( x = 1 ) and ( x = -\frac{2}{\sqrt{3}} ), combined with continuity, guarantees at least one real root in ( \left( -\frac{2}{\sqrt{3}}, 1 \right) ). With ( g(1) < 0 ) and ( g(2) > 0 ), another root is confirmed in ( (1, 2) ). Thus, both sign changes—near ( x = -\frac{2}{\sqrt{3}} ) and in ( (1, 2) ) — anchor the presence and location of real roots through logical and graphical insight."]