If α and β are the zeroes of the quadratic polynomial f(x) = ax² + bx + c, find the value of (1/α − 1/β)

Video Explanation

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Solution

Given polynomial:

f(x) = ax² + bx + c

Let α and β be the zeroes of the given quadratic polynomial.

Step 1: Write the Known Relations

For a quadratic polynomial ax² + bx + c:

α + β = −b/a

αβ = c/a

Step 2: Find the Required Value

1/α − 1/β

= (β − α)/αβ

= −(α − β)/αβ

Step 3: Use the Identity for (α − β)

(α − β)² = (α + β)² − 4αβ

= (−b/a)² − 4(c/a)

= (b² − 4ac)/a²

∴ α − β = √(b² − 4ac)/a

Step 4: Final Calculation

1/α − 1/β

= − [ √(b² − 4ac)/a ] ÷ (c/a)

= − √(b² − 4ac) / c

Final Answer

The required value is − √(b² − 4ac) / c.

Conclusion

Thus, using the relationship between zeroes and coefficients of the quadratic polynomial, the value of (1/α − 1/β) is − √(b² − 4ac) / c.