Question

Quanto é: (x - 1)² = - 16?

Answer 1

Quadrado da diferença de dois termos:

$\boxed{\boxed{(a-b)^{2}=a^{2}-2ab+b^{2}}}$

Unidade imaginária dos números complexos:

$\boxed{\boxed{i^{2}=-1}}~~~(usualmente,~denotamos~i=\sqrt{-1})$
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$(x-1)^{2}=-16\\\\x^{2}-2\cdot x\cdot1+1^{2}=-16\\\\x^{2}-2x+1=-16\\\\x^{2}-2x+1+16=0\\\\x^{2}-2x+17=0~~(a=1~~b=-2~~c=17)$

Resolvendo por Bhaskara:

$\Delta=b^{2}-4ac\\\\\Delta=(-2)^{2}-4\cdot1\cdot17\\\\\Delta=4-68\\\\\Delta=-64$

Delta negativo implica numa equação sem solução real. Portanto, as duas soluções são complexas

$\Delta=(-1)\cdot64\\\\\sqrt{\Delta}=\sqrt{-1}\cdot\sqrt{64}\\\\\sqrt{\Delta}=i\cdot8\\\\\boxed{\boxed{\sqrt{\Delta}=8i}}\\\\\\x=\dfrac{-b\pm\sqrt{\Delta}}{2a}=\dfrac{-(-2)\pm8i}{2\cdot1}=\dfrac{2\pm8i}{2}=\dfrac{2\cdot(1\pm4i)}{2}=1\pm4i$

Logo, as raízes da equação são:

$\boxed{\boxed{x'=1+4i}}\\\\\\\boxed{\boxed{x''=1-4i}}$