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# HELP!!!

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Please help with this equation my 4th grade math teacher is giving extension and i know quadratics and other stuff but i don't know how to solve this properly The real numbers $x$ and $y$ satisfy \begin{align*} x + y &= 4, \\ x^2 + y^2 &= 22, \\ x^4 &= y^4 - 176 \sqrt{7}. \end{align*}Compute $x - y.$

Mar 5, 2020

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and i know quadratics and other stuff but i don't know how to solve this properly
The real numbers $$x$$ and $$y$$ satisfy
\begin{align*} x + y &= 4, \\ x^2 + y^2 &= 22, \\ x^4 &= y^4 - 176 \sqrt{7}. \end{align*}
Compute $$x - y$$.

$$\begin{array}{|rcll|} \hline x^4 &=& y^4 - 176 \sqrt{7} \quad | \quad \times(-1) \\ -x^4 &=& -y^4 + 176 \sqrt{7} \quad | \quad +y^4 \\ y^4-x^4 &=& 176 \sqrt{7} \\ && \boxed{\text{Formula }~a^2-b^2 =(a-b)(a+b) }\\ (y^2-x^2)(y^2+x^2) &=& 176 \sqrt{7} \quad | \quad y^2-x^2=(y-x)(y+x) \\ (y-x)(y+x)(y^2+x^2) &=& 176 \sqrt{7} \quad | \quad y+x = 4 \\ 4(y-x)(y^2+x^2) &=& 176 \sqrt{7} \quad | \quad y^2+x^2 = 22 \\ 4*22(y-x)&=& 176 \sqrt{7} \\ 88(y-x)&=& 176 \sqrt{7} \quad | \quad : 88 \\ y-x &=& 2 \sqrt{7} \quad | \quad \times(-1) \\ \mathbf{ x-y } &=& \mathbf{-2 \sqrt{7}} \\ \hline \end{array}$$

Mar 5, 2020