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: Understanding the Question:
The topic of this question is Polynomials.
For any quadratic polynomial of the form \(ax^2 + bx + c\), its zeroes \(\alpha\) and \(\beta\) are related to its coefficients.
We are asked to find the numerical value of the symmetric rational expression \(\frac{\alpha}{\beta} + \frac{\beta}{\alpha}\).
Instead of calculating the values of \(\alpha\) and \(\beta\) individually, we can rewrite the expression in terms of the sum and product of the zeroes.

Step 2: Key Formula or Approach:
1. The relationship between zeroes and coefficients:
- Sum of zeroes:
\[ \alpha + \beta = -\frac{b}{a} \] - Product of zeroes:
\[ \alpha\beta = \frac{c}{a} \] 2. Rewrite the required algebraic expression using a common denominator:
\[ \frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\alpha^2 + \beta^2}{\alpha\beta} \] 3. Use the algebraic identity for the sum of squares:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \] This allows us to express the target ratio entirely using \(\alpha + \beta\) and \(\alpha\beta\):
\[ \frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{(\alpha + \beta)^2 - 2\alpha\beta}{\alpha\beta} \]

Step 3: Detailed Explanation:

• Identify the coefficients from the given polynomial \(5x^2 - 16x - 10\):
Here, \(a = 5\), \(b = -16\), and \(c = -10\).

• Calculate the sum of the zeroes (\(\alpha + \beta\)):
\[ \alpha + \beta = -\frac{b}{a} = -\frac{-16}{5} = \frac{16}{5} \]

• Calculate the product of the zeroes (\(\alpha\beta\)):
\[ \alpha\beta = \frac{c}{a} = \frac{-10}{5} = -2 \]

• Substitute these values into the rewritten expression for the sum of squares:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \] \[ \alpha^2 + \beta^2 = \left(\frac{16}{5}\right)^2 - 2(-2) \] \[ \alpha^2 + \beta^2 = \frac{256}{25} + 4 \]

• Combine the terms using a common denominator:
\[ \alpha^2 + \beta^2 = \frac{256 + 100}{25} = \frac{356}{25} \]

• Calculate the final value of the required rational expression:
\[ \frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\alpha^2 + \beta^2}{\alpha\beta} \] \[ = \frac{\frac{356}{25}}{-2} \] \[ = -\frac{356}{50} = -\frac{178}{25} \] This can also be written in decimal form as \(-7.12\).


Step 4: Final Answer:
The value of \(\frac{\alpha}{\beta} + \frac{\beta}{\alpha}\) is \(-\frac{178}{25}\).