Question

\(\alpha\) and \(\beta\) are the zeroes of the polynomial \(5x^2 - 16x - 10\). Find the value of \(\frac{\alpha}{\beta} + \frac{\beta}{\alpha}\).

Hint:

Never waste time trying to solve the quadratic equation using the quadratic formula to find the decimal values of \(\alpha\) and \(\beta\).
For almost all symmetric expressions in \(\alpha\) and \(\beta\), you can express the equation cleanly using the sum and product formulas, which saves time and prevents calculation errors!

Answer 1

Step 1: Rewrite the target expression.
\[ \frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\alpha^2+\beta^2}{\alpha\beta} \]
Step 2: Get alpha squared plus beta squared from the discriminant instead.
For \(5x^2-16x-10\), the discriminant is \(D = (-16)^2 - 4(5)(-10) = 256+200 = 456\), and \((\alpha-\beta)^2 = \frac{D}{a^2} = \frac{456}{25}\). Since \(\alpha^2+\beta^2 = (\alpha-\beta)^2 + 2\alpha\beta\), and \(\alpha\beta = \frac{c}{a} = \frac{-10}{5} = -2\): \[ \alpha^2+\beta^2 = \frac{456}{25} + 2(-2) = \frac{456}{25} - 4 = \frac{356}{25} \]
Step 3: Divide by alpha beta.
\[ \frac{\alpha}{\beta}+\frac{\beta}{\alpha} = \frac{\frac{356}{25}}{-2} = -\frac{356}{50} \]
Step 4: Simplify to lowest terms.
\[ -\frac{356}{50} = -\frac{178}{25} \]
\[ \boxed{\frac{\alpha}{\beta}+\frac{\beta}{\alpha} = -\frac{178}{25}} \]