(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 equati...
Question 1 Report
(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.
Answer Details
(a) \(\dfrac{5y - x}{8y + 3x} = \dfrac15\). Cross-multiplying:
\[5(5y - x) = 8y + 3x \;\Rightarrow\; 25y - 5x = 8y + 3x\]
\[17y = 8x \;\Rightarrow\; \frac{x}{y} = \frac{17}{8} = 2.13 \text{ (2 d.p.)}\]
(b) Since \(3\) is a root of \(x^2 + bx - 15 = 0\):
\[3^2 + 3b - 15 = 0 \;\Rightarrow\; 9 + 3b - 15 = 0 \;\Rightarrow\; 3b = 6 \;\Rightarrow\; b = 2\]
The equation becomes \(x^2 + 2x - 15 = 0 = (x + 5)(x - 3)\). Hence the other root is \(x = -5\).
(Check by product of roots: \(3 \times (-5) = -15\), matching the constant term.)
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