(a) Given that \(\frac{5y - x}{8y + 3x} = \frac{1}{5}\), find the value of \(\frac{x}{y}\) to two decimal places. (b) If 3 is a root of the quadratic equation \(x^{2} + bx - 15 = 0\), determine the value of b. Find the other root.
Explanation
(a) \(\frac{5y - x}{8y + 3x} = \frac{1}{5}\) \(5(5y - x) = 8y + 3x \implies 25y - 5x = 8y + 3x\) \(25y - 8y = 3x + 5x \implies 17y = 8x\) \(\therefore \frac{x}{y} = \frac{17}{8} = 2.125 \approxeq 2.13\) (b) \(x^{2} + bx - 15 = 0\) Since x = 3 is a root of the equation, f(3) = 0. \(3^{2} + 3b - 15 = 0 \implies 3b = 6\) \(b = 2\) \(\therefore\) The equation is \(x^{2} + 2x - 15 = 0\) \(x^{2} - 3x + 5x - 15 = 0 \implies x(x - 3) + 5(x - 3) = 0\) \((x - 3)(x + 5) = 0\) The second root of the equation is x = -5.