Functions, 2ADV EQ-Bank 28 Given \(p\) and \(q\) are rational numbers, and \(p, q \neq 0\), show \(px^2-(p+q) x+q=0\) has rational roots. (3 marks) --- 10 WORK AREA LINES (style=lined) --- Show Answers Only \(\text{Proof (See Worked Solution)}\) Show Worked Solution \(\Delta\) \(=b^2-4 a c\) \(=[-(p+q)]^2-4 \times p \times q\) \(=p^2+2 p q+q^2-4 p q\) \(=p^2-2 p q+q^2\) \(=(p-q)^2\) \(\text{Roots of equation using quadratic formula:}\) \(x\) \(=\dfrac{(p+q) \pm \sqrt{(p-q)^2}}{2 p}\) \(=\dfrac{p+q+(p-q)}{2 p} \ \ \text{or} \ \ \dfrac{p+q-(p-q)}{2 p}\) \(=1 \ \ \text{or} \ \ \dfrac{q}{p}\). \(\text{Since \(p, q\) are rational, all roots are rational.}\)