Question

Escreva as equações do 2° grau na forma reduzida e indica quantas raízes reais elas possuem:
a) 5 – 11x2 = - 8x
b) 4 + 3x = - x2 + 2
c) x (x + 2) = -5

Answer 1

Resposta:

A) $x=-\frac{-4+\sqrt{71}}{11},\:x=\frac{4+\sqrt{71}}{11}$

B) $x=-2,\:x=-1$

C) $x=-1+2i,\:x=-1-2i$

Explicação passo a passo:

A) $5-11x^2=-8x$

$5-11x^2+8x=-8x+8x$

$-11x^2+8x+5=0$

$x_{1,\:2}=\frac{-8\pm \sqrt{8^2-4\left(-11\right)\cdot \:5}}{2\left(-11\right)}$

$x_{1,\:2}=\frac{-8\pm \:2\sqrt{71}}{2\left(-11\right)}$

$x_1=\frac{-8+2\sqrt{71}}{2\left(-11\right)},\:x_2=\frac{-8-2\sqrt{71}}{2\left(-11\right)}$

$x=-\frac{-4+\sqrt{71}}{11},\:x=\frac{4+\sqrt{71}}{11}$

B) $4+3x=-x^2+2$

$-x^2+2=4+3x$

$-x^2+2-3x=4+3x-3x$

$-x^2+2-3x=4$

$-x^2+2-3x-4=4-4$

$-x^2-3x-2=0$

$x_{1,\:2}=\frac{-\left(-3\right)\pm \sqrt{\left(-3\right)^2-4\left(-1\right)\left(-2\right)}}{2\left(-1\right)}$

$x_{1,\:2}=\frac{-\left(-3\right)\pm \:1}{2\left(-1\right)}$

$x_1=\frac{-\left(-3\right)+1}{2\left(-1\right)},\:x_2=\frac{-\left(-3\right)-1}{2\left(-1\right)}$

$x=-2,\:x=-1$

C) $x\left(x+2\right)=-5$

$x^2+2x=-5$

$x^2+2x+5=-5+5$

$x^2+2x+5=0$

$x_{1,\:2}=\frac{-2\pm \sqrt{2^2-4\cdot \:1\cdot \:5}}{2\cdot \:1}$

$x_{1,\:2}=\frac{-2\pm \:4i}{2\cdot \:1}$

$x_1=\frac{-2+4i}{2\cdot \:1},\:x_2=\frac{-2-4i}{2\cdot \:1}$

$x=-1+2i,\:x=-1-2i$