Question

Simplifique a expressão algébrica abaixo:
(+ )²+ ( + )² ( − ) + ( − )³ + √+√x +$\frac{x}⁴{x}⁴$ = 24

Answer 1

$(x+4)^2+(x+3)^2(x-3)+(x-2)^3+\sqrt{169}\,x+\sqrt{169}\,x+\frac{x^4}{x^4}~=~24$

Vamos desenvolver alguns dos termos separadamente para facilitar a apresentação.

$\rightarrow~\underline{(x+4)^2}\,=\\\\\\~~~~~~=~(x+4).(x+4)\\\\\\~~~~~~=~x\,.\,x+x\,.\,4+4\,.\,x+4\,.\,4\\\\\\~~~~~~=~x^2+4x+4x+16\\\\\\~~~~~~=~\boxed{x^2+8x+16}$

$\rightarrow~\underline{(x+3)^2}\,=\\\\\\~~~~~~=~(x+3).(x+3)\\\\\\~~~~~~=~x\,.\,x+x\,.\,3+3\,.\,x+3\,.\,3\\\\\\~~~~~~=~x^2+3x+3x+9\\\\\\~~~~~~=~\boxed{x^2+6x+9}$

$\rightarrow~\underline{(x+3)^2(x-3)}\,=\\\\\\~~~~~~=~(x^2+6x+9).(x-3)\\\\\\~~~~~~=~x^2\,.\,x+x^2\,.\,(-3)+6x\,.\,x+6x\,.\,(-3)+9\,.\,x+9\,.\,(-3)\\\\\\~~~~~~=~x^3-3x^2+6x^2-18x+9x-27\\\\\\~~~~~~=~\boxed{x^3+3x^2-9x-27}$

$\rightarrow~\underline{(x-2)^3}\,=\\\\\\~~~~~~=~(x-2).(x-2).(x-2)\\\\\\~~~~~~=~(\,x\,.\,x+x\,.\,(-2)+(-2)\,.\,x+(-2)\,.\,(-2)\,)~.~(x-2)\\\\\\~~~~~~=~(x^2-4x+4)\,.\,(x-2)\\\\\\~~~~~~=~x^2\,.\,x+x^2\,.\,(-2)+(-4x)\,.\,x+(-4x)\,.\,(-2)+4\,.\,x+4\,.\,(-2)\\\\\\~~~~~~=~x^3-2x^2-4x^2+8x+4x-8\\\\\\~~~~~~=~\boxed{x^3-6x^2+12x-8}$

Vamos agora passar substituir os termos desenvolvidos na equação dada:

$(x+4)^2+(x+3)^2(x-3)+(x-2)^3+\sqrt{169}\,x+\sqrt{169}\,x+\frac{x^4}{x^4}~=~24\\\\\\\\(x^2+8x+16)+(x^3+3x^2-9x-27)+(x^3-6x^2+12x-8)+\underbrace{13x+13x}_{26x}+\underbrace{\frac{x\!\!\!\backslash^4}{x\!\!\!\backslash^4}}_{1}~=~24\\\\\\x^2+8x+16+x^3+3x^2-9x-27+x^3-6x^2+12x-8+26x+1~=~24\\\\\\\\Associando~os~termos~semelhantes\\\\\\\\(x^3+x^3)+(x^2+3x^2-6x^2)+(8x-9x+12x+26x)+(16-27-8+1-24)~=~0$

$\boxed{2x^3-2x^2+37x-42~=~0}$

Observe que a questão não possui gabarito correto. Por favor, verifique se a expressão e as alternativas foram corretamente digitadas.