Question

1- escreva as equações na forma reduzida e verifique se as mesmas tem raízes reais.
A - x.(x+1)=2.(x+3)
B - (x+2)^2-4=3x.(x+1)

Answer 1

Resposta:

$x\left(x+1\right)=2\left(x+3\right)\\\\x^2+x=2x+6\\\\x^2+x-6=2x+6-6\\\\x^2+x-6=2x\\\\x^2+x-6-2x=2x-2x\\\\x^2-x-6=0\\\\x_{1,\:2}=\frac{-\left(-1\right)\pm \sqrt{\left(-1\right)^2-4\cdot \:1\cdot \left(-6\right)}}{2\cdot \:1}\\\\x_{1,\:2}=\frac{-\left(-1\right)\pm \:5}{2\cdot \:1}\\\\x_1=\frac{-\left(-1\right)+5}{2\cdot \:1},\:x_2=\frac{-\left(-1\right)-5}{2\cdot \:1}\\\\x=3,\:x=-2$

_______________________________________________________

$\left(x+2\right)^2-4=3x\left(x+1\right)\\\\x^2+4x=3x^2+3x\\\\x^2+4x-3x=3x^2+3x-3x\\\\x^2+x=3x^2\\\\x^2+x-3x^2=3x^2-3x^2\\\\-2x^2+x=0\\\\x_{1,\:2}=\frac{-1\pm \sqrt{1^2-4\left(-2\right)\cdot \:0}}{2\left(-2\right)}\\\\x_{1,\:2}=\frac{-1\pm \:1}{2\left(-2\right)}\\\\x_1=\frac{-1+1}{2\left(-2\right)},\:x_2=\frac{-1-1}{2\left(-2\right)}\\\\x=0,\:x=\frac{1}{2}$

Answer 2

Resposta:

A - x.(x+1)=2.(x+3)

x²+x=2x+6

x²-x-6=0

Δ=1²-4*1*(-6)=25 >0 ==>2 raízes Reais

x'=[1+√(1+24)]/2=(1+5)/2=3

x''=[1-√(1+24)]/2=(1-5)/2=-2

B - (x+2)^2-4=3x.(x+1)

x²+4x+4-4=3x²+3x

-2x²+x=0

Δ=1-4*(-2)*0=1 ==>==>2 raízes Reais

x*(-2x+1)=0

x=0

ou

-2x+1=0 ==>x=1/2