If α and β are the roots of the equation 3x2 + 5x - 2 = 0, find the value of 1/α + 1/β

Analyze the question: We need 1/α + 1/β for the roots of 3x² + 5x - 2 = 0. We don't need to find the roots individually. We can use the sum and product of roots.

Concept: For ax² + bx + c = 0:

• Sum of roots: α + β = -b/a
• Product of roots: αβ = c/a

Also, 1/α + 1/β = (α + β)/(αβ).

Solution:

Here a = 3, b = 5, c = -2.

• α + β = -5/3
• αβ = -2/3

Then:

1/α + 1/β = (α + β)/(αβ) = (-5/3) ÷ (-2/3) = (-5/3) × (-3/2) = 5/2

Check by finding the roots: 3x² + 5x - 2 = (3x - 1)(x + 2) = 0, so x = 1/3 or x = -2. 1/α + 1/β = 3 + (-1/2) = 5/2 ✓

Answer: 5/2, which is option D.

Tip: For any quadratic, 1/α + 1/β = -b/c. Here that is -5/(-2) = 5/2. Watch out for the signs, since two negatives give a positive. Option A is just the sum of the roots, a common trap.