Question

Cho x-y=4 và x² -y²=86. Tính x³-y³

Answer 1

\(x-y=4\Rightarrow\left(x-y\right)^2=16\Leftrightarrow x^2-2xy+y^2=16\)

\(\Leftrightarrow86-2xy=16\Leftrightarrow2xy=70\Leftrightarrow xy=35\)

\(x^2-y^2=86\Leftrightarrow\left(x-y\right)\left(x+y\right)=86\Leftrightarrow4 \left(x+y\right)=86\Leftrightarrow x+y=\frac{43}{2}\)

\(\Rightarrow\left(x+y\right)^2=\frac{1849}{4}\Leftrightarrow x^2+2xy+y^2=\frac{1849}{4}\Leftrightarrow x^2+2.35+y^2=\frac{1849}{4}\)

\(\Leftrightarrow x^2+70+y^2=\frac{1849}{4}\Leftrightarrow x^2+y^2=\frac{1569}{4}\)

\(x^3-y^3=\left(x-y\right)\left(x^2+xy+y^2\right)=4.\left(\frac{1569}{4}+35\right)=4.\frac{1709}{4}=1709\)

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