If α and β are the zeroes of the quadratic polynomial f(x) = ax² + bx + c, find the value of 1/(aα + b) + 1/(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: Simplify the Given Expression

1/(aα + b) + 1/(aβ + b)

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

= [a(α + β) + 2b] / [a²αβ + ab(α + β) + b²]

Step 3: Substitute the Values

Numerator:

a(α + β) + 2b = a(−b/a) + 2b = −b + 2b = b

Denominator:

a²αβ + ab(α + β) + b²

= a²(c/a) + ab(−b/a) + b²

= ac − b² + b² = ac

Step 4: Find the Required Value

1/(aα + b) + 1/(aβ + b)

= b/(ac)

Final Answer

The required value is b/(ac).

Conclusion

Thus, using the relationship between zeroes and coefficients of the quadratic polynomial, the value of 1/(aα + b) + 1/(aβ + b) is b/(ac).