The real root of the equation $x^{3}+12x-12=0$, is:
- $2\sqrt[3]{2}-\sqrt[3]{5}$
- $2\sqrt[3]{2}+\sqrt[3]{5}$
- $2\sqrt[3]{2}+\sqrt[3]{4}$
- $2\sqrt[3]{2}-\sqrt[3]{4}$
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Correct answer
Explanation
Using Cardano's method for x^3 + 12x - 12 = 0, let x = u + v. The solution involves cube roots. The expression 2*cbrt(2) - cbrt(4) is the correct real root.
AI explanation
Using Cardano's formula for the depressed cubic x^3 + px + q = 0, we identify p = 12 and q = -12. We calculate the discriminant value of (-q/2)^2 + (p/3)^3, which is 6^2 + 4^3 = 36 + 64 = 100. The real root is given by the sum of the cube roots: cube_root(6 + sqrt(100)) + cube_root(6 - sqrt(100)). This simplifies to cube_root(16) - cube_root(4), which can be factored as 2(cube_root(2)) - cube_root(4).