If \(x^2 + \frac{1}{x^2} = 23\), then the value of \(\frac{4x}{5x^2 - 3x + 5}\) is:
\(\frac{2}{11}\)
Given \(x^2 + \frac{1}{x^2} = 23\).
Add 2 to both sides: \(x^2 + 2 + \frac{1}{x^2} = 25\), i.e. \(\left(x + \frac{1}{x}\right)^2 = 25\), so \(x + \frac{1}{x} = 5\).
Divide the numerator and denominator of the expression by x: \(\frac{4x}{5x^2 - 3x + 5} = \frac{4}{5x - 3 + \frac{5}{x}} = \frac{4}{5\left(x + \frac{1}{x}\right) - 3}\).
Substitute: \(\frac{4}{5 \times 5 - 3} = \frac{4}{22} = \frac{2}{11}\).
Hence, the answer is \(\frac{2}{11}\).
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