If α and β are the zeroes of the quadratic polynomial f(x) = ax² + bx + c, find the value of (α²β − αβ²)

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: Simplify the Given Expression

α²β − αβ²

= αβ(α − β)

Step 3: Find (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: Find the Required Value

α²β − αβ²

= (c/a) × [√(b² − 4ac)/a]

= c√(b² − 4ac)/a²

Final Answer

The required value is c√(b² − 4ac) / a².

Conclusion

Thus, using the relationship between zeroes and coefficients of the quadratic polynomial, the value of (α²β − αβ²) is c√(b² − 4ac) / a².