Question

Sendo x1 e x2 raízes da equação 2x²-5x-1=0, calcule o valor de x1²+x2²

Answer 1

$2x^2 - 5x - 1 = 0 \\\\ \Delta = b^2-4ac \\ \Delta = (-5)^2 - 4.2.(-1) \\ \Delta = 25 + 8 \\ \Delta = 33 \\\\ x^+ = \frac{-b+ \sqrt{\Delta}}{2a} \\\\ x^+ = \frac{-(-5) + \sqrt{33} }{2.2}\\\\ x^+ = \frac{5 + \sqrt{33} }{4} ~ \qquad \qquad\qquad x^-= \frac{5 - \sqrt{33} }{4} \\\\\\ (x^+)^2 + (x^-)^2 \\\\ (\frac{5 + \sqrt{33} }{4})^2 + (\frac{5 - \sqrt{33} }{4})^2 \\\\ \frac{5^2 + 2.5. \sqrt{33} + \sqrt{33}^2 } {4} + \frac{5^2 - 2.5. \sqrt{33} + \sqrt{33}^2 } {4}$

$\frac{25 + 10 \sqrt{33}+33 +25 -10 \sqrt{33} + 33 } {4} \\\\ \frac{25 + 25 + 33 +33 + 10 \sqrt{33}-10 \sqrt{33}} {4} \\\\ \frac {116} {4} = \boxed{29}$