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

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: Combine the Given Expression

β/(aα + b) + α/(aβ + b)

= [β(aβ + b) + α(aα + b)] / [(aα + b)(aβ + b)]

= [a(α² + β²) + b(α + β)] / [a²αβ + ab(α + β) + b²]

Step 3: Substitute the Values

α² + β² = (α + β)² − 2αβ

= (−b/a)² − 2(c/a)

= b²/a² − 2c/a

Numerator:

a(α² + β²) + b(α + β)

= a(b²/a² − 2c/a) + b(−b/a)

= b²/a − 2c − b²/a

= −2c

Denominator:

a²αβ + ab(α + β) + b²

= a²(c/a) + ab(−b/a) + b²

= ac − b² + b²

= ac

Step 4: Find the Required Value

β/(aα + b) + α/(aβ + b)

= (−2c)/(ac)

= −2/a

Final Answer

The required value is −2/a.

Conclusion

Thus, using the relationship between zeroes and coefficients of the quadratic polynomial, the value of β/(aα + b) + α/(aβ + b) is −2/a.