Question

If \(\alpha\) and \(\beta\) are roots of the quadratic equation \(3x^2 - 5x + 2 = 0\) then the value of \(\alpha^2 + \beta^2\) is

• A. \(\frac{13}{9}\)
• B. \(\frac{9}{13}\)
• C. \(\frac{5}{3}\)
• D. \(\frac{3}{5}\)

Hint:

Remember the key identity \(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta\). It is frequently used in problems involving the sum of the squares of roots.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
For a quadratic equation \(ax^2 + bx + c = 0\), the sum of the roots (\(\alpha + \beta\)) is \(-b/a\) and the product of the roots (\(\alpha\beta\)) is \(c/a\). We can use these relationships to find the value of expressions involving the roots, like \(\alpha^2 + \beta^2\).

Step 2: Key Formula or Approach:
1. Find the sum and product of the roots.
2. Use the algebraic identity: \(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta\).

Step 3: Detailed Explanation:
For the equation \(3x^2 - 5x + 2 = 0\), we have \(a=3, b=-5, c=2\).
Sum of roots:
\[ \alpha + \beta = -\frac{b}{a} = -\frac{-5}{3} = \frac{5}{3} \] Product of roots:
\[ \alpha\beta = \frac{c}{a} = \frac{2}{3} \] Now, use the identity to find \(\alpha^2 + \beta^2\):
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \] \[ = \left(\frac{5}{3}\right)^2 - 2\left(\frac{2}{3}\right) \] \[ = \frac{25}{9} - \frac{4}{3} \] To subtract, find a common denominator (9):
\[ = \frac{25}{9} - \frac{12}{9} = \frac{13}{9} \]

Step 4: Final Answer:
The value of \(\alpha^2 + \beta^2\) is \(\frac{13}{9}\).