Question

gente urgente prof passou esse desafio. me ajudem por favor.

Answer 1

$x^{2}-4x-7=0\;\;\Rightarrow\;\;a=1,\;\;b=-4,\;\;c=-7\\ \\ \\ \Delta=b^{2}-4ac\\ \\ \Delta=\left(-4 \right )^{2}-4\cdot \left(1 \right )\cdot \left(-7 \right )\\ \\ \Delta=16+28\\ \\ \Delta=44\\ \\ \Delta=2^{2}\cdot 11\\ \\ \\ x=\dfrac{-b \pm \sqrt{\Delta}}{2a}\\ \\ x=\dfrac{-\left(-4 \right ) \pm \sqrt{2^{2}\cdot 11}}{2\cdot \left(1 \right )}\\ \\ x=\dfrac{4 \pm 2\sqrt{11}}{2}\\ \\ x=\dfrac{2\cdot \left(2 \pm \sqrt{11} \right )}{2}\\ \\ x=2 \pm \sqrt{11}\\ \\ x_{1}=2+\sqrt{11},\;\;\;x_{2}=2-\sqrt{11}$


Sendo assim, temos que

$x_{1}^{2}+x_{2}^{2}=\left(2+\sqrt{11} \right )^{2}+\left(2-\sqrt{11} \right )^{2}\\ \\ x_{1}^{2}+x_{2}^{2}=\left(2^{2}+2\cdot 2\cdot \sqrt{11}+\sqrt{11}^{2} \right )+\left(2^{2}-2\cdot 2\cdot \sqrt{11}+\sqrt{11}^{2} \right )\\ \\ x_{1}^{2}+x_{2}^{2}=\left(4+4\sqrt{11}+11 \right )+\left(4-4\sqrt{11}+11 \right )\\ \\ x_{1}^{2}+x_{2}^{2}=\left(15+4\sqrt{11} \right )+\left(15-4\sqrt{11} \right )\\ \\ x_{1}^{2}+x_{2}^{2}=15+15+4\sqrt{11}-4\sqrt{11}\\ \\ \boxed{x_{1}^{2}+x_{2}^{2}=30}$