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If A and B are the roots of x^2 - 4x + 1 = 0, then find A^3 + B^3.

Jun 21, 2020

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This is simple. x^2-4x+1 is not factorable so we plug it into the quadratic equation.

$$x = \frac{-b \pm \sqrt{b^2-4ac}} {2a}\\ x=\frac{4\pm\sqrt{(-4)^2-4(1)(1)}}{2(1)}\\ x=\frac{4\pm\sqrt{12}}{2}\\ x=\frac{4\pm2\sqrt{3}}{2}\\ x=2\pm\sqrt{3}$$

Cube the two roots.

$$(2+\sqrt{3})^3+(2-\sqrt{3})^3\\ =\boxed{52}$$

Jun 21, 2020