Question

Determine a raiz real da equação"(x + 5)² - (2x+1)² = 2. Como se resolve essa equação do segundo grau?

Answer 1

$(x + 5)^2 - (2x+1)^2= 2\to \\\\ ( x^{2} +10x+25)-(4 x^{2} +4x+1)=2\to \\\\ x^{2} +10x+25-4 x^{2} -4x-1-2=0\to \\\\ -3 x^{2} +6x+22=0\\\\ a=-3;\ b=6;\ c=22\\\\\\ \Delta=b^2-4ac\to \Delta=6^2-4.(-3).22\to \Delta=36+264\to \boxed{\Delta=300}\\\\ x' \neq x''$



$x= \dfrac{-b\pm \sqrt{\Delta} }{2a} \to ~~ x= \dfrac{-6\pm \sqrt{3'00} }{2.(-3)} \to ~~ x= \dfrac{-6\pm 10\sqrt{3} }{-6} \to \\\\\\ x'= \dfrac{-6+ 10\sqrt{3} }{-6} \to ~~ \boxed{x'= \dfrac{3-5\sqrt{3} }{3} }\\\\\\ x'= \dfrac{-6-10\sqrt{3} }{-6} \to ~~ \boxed{x'= \dfrac{3+5\sqrt{3} }{3} }\\\\\\\\\\ \boxed{\boxed{S=\{ \dfrac{3-5\sqrt{3} }{3} ~~;~~ \dfrac{3+5\sqrt{3} }{3} \}}}$