Question

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1) A soma de um número com o seu quadrado é 90. Calcule esse numero. (R: 9 e -10)

2) A soma do quadrado de um número com o próprio número é 12. Calcule esse numero. (R: 3 e -4)

Answer 1

Equações do 2° grau: Utilize Bhaskara:

$\Delta=b^2-4ac\\\\\\ x= \dfrac{-b\pm \sqrt{ \Delta} }{2a}$

$1)~~x+x^2=90\to \\\\x^2+x-90=0\\\\ a=1;b=1;c=-90\\\\\\ \Delta=1^2-4.1.(-90)\to \Delta=1+360\to \boxed{\Delta=361}\\\\x' \neq x''\\\\\\ x= \dfrac{-1\pm \sqrt{361} }{2.1} \to ~~x= \dfrac{-1\pm 19 }{2} \to \\\\\\ x'= \dfrac{-1+ 19 }{2} \to~~ x'= \dfrac{18 }{2} \to ~~ \boxed{x'=9}\\\\\\ x''= \dfrac{-1-19 }{2} \to~~ x''= \dfrac{-20 }{2} \to ~~ \boxed{x''=-10}\\\\\\\\\\ \boxed{\boxed{S=\{-10~;~9\}}}$






$2)~~x^2+x=12\to\\\\ x^2+x-12=0\\\\ a=1;b=1;c=-12\\\\\\ \Delta=1^2-4.1.(-12)\to \Delta=1+48\to \boxed{\Delta=49}\\\\x' \neq x''\\\\\\ x= \dfrac{-1\pm \sqrt{ 49} }{2.1} \to ~~x= \dfrac{-1\pm 7}{2} \to \\\\\\ x'= \dfrac{-1+ 7 }{2} \to ~~x'= \dfrac{6 }{2} \to ~~\boxed{x'=3}\\\\\\ x''= \dfrac{-1- 7 }{2} \to ~~x''= \dfrac{-8 }{2} \to ~~\boxed{x''=-4}\\\\\\\\\\ \boxed{\boxed{S=\{-4~;~3\}}}$