Question

resolva as equações do 2 grau em IR

a) x²=7x-10
b) x(x+1)=30

Answer 1

$a) x^{2} =7x-10\\\\ x^{2} -7x+10=0\\\\ a=1;b=-7;c=10\\\\ \Delta=b^2-4ac\to \Delta=(-7)^2-4*1*10\to \Delta=49-40\to \Delta=9\\\\ x' \neq x''\\\\x= \frac{-b\pm \sqrt{\Delta} }{2a} \to x= \frac{-(-7)\pm \sqrt{9} }{2*1} \to x= \frac{7\pm 3 }{2} \to\\\\ x'= \frac{7+3 }{2} \to x'= \frac{10 }{2} \to x'=5\\\\ x''= \frac{7-3 }{2} \to x''= \frac{4 }{2} \to x''=2\\\\\\ S=(2;5)$





$b) x(x+1)=30\\\\ x^{2} +x-30=0\\\\ a=1;b=1;c=-30\\\\ \Delta=b^2-4ac\to \Delta=1^2-4*1*(-30)\to \Delta=1+120\to \Delta=121\\\\ x' \neq x''\\\\x= \frac{-b\pm \sqrt{\Delta} }{2a} \to x= \frac{-1\pm \sqrt{121} }{2*1} \to x= \frac{-1\pm 11 }{2} \to\\\\x'= \frac{-1+11 }{2} \to x'= \frac{10 }{2} \to x'=5\\\\ x''= \frac{-1-11 }{2} \to x''= \frac{-12 }{2} \to x''=-6\\\\\\ S=(-6;5)$

Answer 2

$x^{2} = 7x - 10 \\ x^{2} - 7x = -10 \\ x^{2} - 7x +10=0$

Δ = $(-7)^{2} -4 *10$
Δ = $(-7)^{2} - 40$
Δ= $49-40$
Δ = $9$

$x_{1,2} = \frac{7± \sqrt{9} }{2}$
$x_{1,2} = \frac{7±3}{2}$

$x_{1} = \frac{7+3}{2} \\ \\ x_{1} = \frac{10}{2} \\ \\ x_{1} =5 \\ \\ \\ x_{2} = \frac{7-3}{2} \\ \\ x_{2} = \frac{4}{2} \\ \\ x_{2} = 2$

b) x(x+1)=30
$x*(x+1)=30 \\ x*x+x=30 \\ x^{2} +x=30 \\ x^{2}+x-30=0$
Δ = $1^{2} -4(-30)$
Δ=$1^{2} +120$
Δ=$1+120$
Δ=$121$

$x_{1,2} = \frac{(-1)± \sqrt{121} }{2} \\ \\ x_{1,2} = \frac{(-1)±11}{2} \\ \\ x_{1} = \frac{(-1)+11}{2} \\ \\ x_{1} = \frac{10}{2} \\ \\ x_{1} = 5 \\ \\ \\ x_{2} = \frac{-1-11}{2} \\ \\ x_{2} = \frac{-12}{2} \\ \\ x_{2} = -6$