Question

Como faz essa conta?

4x(x+6)x²=5x²

Answer 1

sendo:

4x * (x + 6) * x^2 = 5x^2

x^2 * (4x^2 + 24x - 5) = 0

(4x^2 + 24x - 5) = 0

delta

d = 24^2 + 4*4*5 = 656 = 16 * 41

as raizes

x1 = (-24 + 4*√41)/8 = -3 + √41/2

x2 = (-24 - 4√41)/8 = -3 - √41/2

x3 = 0

Answer 2

$4x\left(x+6\right)x^2=5x^2$

$\mathrm{Subtrair\:}5x^2\mathrm{\:de\:ambos\:os\:lados}\\ 4x\left(x+6\right)x^2-5x^2=5x^2-5x^2\\ \mathrm{Simplificar}\\ 4x\left(x+6\right)x^2-5x^2=0$

$\mathrm{Fatorar\:}4x\left(x+6\right)x^2-5x^2:\quad x^2\left(4x^2+24x-5\right)\\ \mathrm{Expandir}\:4x\left(x+6\right)x^2-5x^2:\quad 4x^4+24x^3-5x^2\\ 4x\left(x+6\right)x^2-5x^2\\ 4x\left(x+6\right)x^2=4x^3\left(x+6\right)\\ =4x^3\left(x+6\right)-5x^2\\ \mathrm{Expandir}\:4x^3\left(x+6\right):\quad 4x^4+24x^3\\ =4x^4+24x^3-5x^2$

$x^2\left(4x^2+24x-5\right)=0\\ Usando\:o\:principio\:do\:fator\:zero:\\ \mathrm{\:Se}\:ab=0\:\mathrm{entao}\:a=0\:\mathrm{ou}\:b=0\:\left(\mathrm{ou\:ambos}\:a=0\:\mathrm{e}\:b=0\right)$

$x=0$

$\mathrm{Resolver\:}\:4x^2+24x-5=0:\quad x=\frac{-6+\sqrt{41}}{2},\:x=-\frac{6+\sqrt{41}}{2}\\ 4x^2+24x-5=0\\ \mathrm{Resolver\:com\:a\:formula\:quadratica}\\ \mathrm{Formula\:geral\:para\:equacoes\:de\:segundo\:grau:}\\ \mathrm{Para\:uma\:equacao\:de\:segundo\:grau\:da\:forma\:}ax^2+bx+c=0\mathrm{\:as\:solucoes\:sao\:}\\ x_{1,\:2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

$\mathrm{Para\:}\quad a=4,\:b=24,\:c=-5:\quad x_{1,\:2}=\frac{-24\pm \sqrt{24^2-4\cdot \:4\left(-5\right)}}{2\cdot \:4}\\ x=\frac{-24+\sqrt{24^2-4\cdot \:4\left(-5\right)}}{2\cdot \:4}:\quad \frac{-6+\sqrt{41}}{2}\\ \frac{-24+\sqrt{24^2-4\cdot \:4\left(-5\right)}}{2\cdot \:4}\\ \mathrm{Aplicar\:a\:regra}\:-\left(-a\right)=a\\ =\frac{-24+\sqrt{24^2+4\cdot \:4\cdot \:5}}{2\cdot \:4}\\ \sqrt{24^2+4\cdot \:4\cdot \:5}=\sqrt{656}\\ \sqrt{24^2+4\cdot \:4\cdot \:5}\\ 24^2=576\\ =\sqrt{576+4\cdot \:4\cdot \:5}$

$\mathrm{Multiplicar\:os\:numeros:}\:4\cdot \:4\cdot \:5=80\\ =\sqrt{576+80}\\ \mathrm{Somar:}\:576+80=656\\ =\sqrt{656}\\ =\frac{-24+\sqrt{656}}{2\cdot \:4}\\ \mathrm{Multiplicar\:os\:numeros:}\:2\cdot \:4=8\\ =\frac{-24+\sqrt{656}}{8}$

$\sqrt{656}=4\sqrt{41}\\ \sqrt{656}\\ \mathrm{Decomposicao\:em\:fatores\:primos\:de\:}656:\quad 2^4\cdot \:41\\ =\sqrt{2^4\cdot \:41}\\ \mathrm{Aplicar\:as\:propriedades\:dos\:radicais}:\quad \sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b}\\ =\sqrt{41}\sqrt{2^4}\\ \mathrm{Aplicar\:as\:propriedades\:dos\:radicais}:\quad \sqrt[n]{a^m}=a^{\frac{m}{n}}\\ \sqrt{2^4}=2^{\frac{4}{2}}=2^2\\ =2^2\sqrt{41}\\ \mathrm{Simplificar}\\ =4\sqrt{41}\\ =\frac{-24+4\sqrt{41}}{8}$

$x=\frac{-24-\sqrt{24^2-4\cdot \:4\left(-5\right)}}{2\cdot \:4}:\quad -\frac{6+\sqrt{41}}{2}\\ \frac{-24-\sqrt{24^2-4\cdot \:4\left(-5\right)}}{2\cdot \:4}\\ \mathrm{Aplicar\:a\:regra}\:-\left(-a\right)=a\\ =\frac{-24-\sqrt{24^2+4\cdot \:4\cdot \:5}}{2\cdot \:4}\\ \sqrt{24^2+4\cdot \:4\cdot \:5}=\sqrt{656}\\ =\frac{-24-\sqrt{656}}{2\cdot \:4}\\ \mathrm{Multiplicar\:os\:numeros:}\:2\cdot \:4=8\\ =\frac{-24-\sqrt{656}}{8}\\ \sqrt{656}=4\sqrt{41}\\ \frac{-24-4\sqrt{41}}{8}$

$\mathrm{Fatorar}\:-24-4\sqrt{41}:\quad -4\left(6+\sqrt{41}\right)\\ =-\frac{4\left(6+\sqrt{41}\right)}{8}\\ \mathrm{Eliminar\:o\:fator\:comum:}\:4\\ =-\frac{6+\sqrt{41}}{2}\\ \mathrm{As\:solucoes\:para\:a\:equacao\:de\:segundo\:grau\:sao:\:}\\ x=\frac{-6+\sqrt{41}}{2},\:x=-\frac{6+\sqrt{41}}{2}\\ \mathrm{As\:solucoes\:sao}\\ x=0,\:x=\frac{-6+\sqrt{41}}{2},\:x=-\frac{6+\sqrt{41}}{2}$

$x=0$