Question

If x - 3/x = 7 find i) x² + 9/x²

ii) x + 3/x
iii) x² - 9/x²
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Answer 1

`\text{Given that,}\\\\x-\frac 3x = 7\\\\\\\textbf{i).}\\\\\\x^2 + \frac 9{x^2}\\\\\\=x^2 + \left(\frac 3 x \right)^2\\\\\\=\left(x- \frac 3x \right)^2 + 2 \cdot x \cdot \frac 3x\\\\\\=7^2 +6\\\\\\=55`

`\textbf{ii)}\\\\~~~~~\left(x+\frac 3 x \right)^2 = \left( x - \frac 3x \right)^2+4 \cdot x \cdot \frac 3x\\\\\\\implies \left(x+\frac 3 x \right)^2 = 7^2 +12\\\\\\\implies \left(x+\frac 3 x \right)^2 =61\\\\\\\implies x+ \frac 3x = \pm√(61)\\\\\\`

`\textbf{iii)}\\\\\\x^2 -\frac 9{x^2}\\\\\\=x^2 - \left(\frac 3{x} \right)^2\\\\\\=\left( x-\frac 3x \right)\left( x+\frac 3x \right)\\\\\\=7√(61)~~ \text{or}~~ -7√(61)`