If α and β are the zeros of the polynomial f(x) = ax² + bx + c, find the value of 1/α² + 1/β²

Video Explanation

Watch the video explanation below:

Given

f(x) = ax² + bx + c

α and β are the zeros of the polynomial.

To Find

The value of 1/α² + 1/β².

Solution

For a quadratic polynomial ax² + bx + c:

α + β = −b/a

αβ = c/a

Step 1: Use the Identity

1/α² + 1/β² = (α² + β²)/(αβ)²

α² + β² = (α + β)² − 2αβ

Step 2: Substitute the Values

α² + β² = (−b/a)² − 2(c/a)

= b²/a² − 2c/a

(αβ)² = (c/a)²

Step 3: Find the Required Value

1/α² + 1/β² = (b²/a² − 2c/a) ÷ (c²/a²)

= (b² − 2ac) / c²

Final Answer

1/α² + 1/β² = (b² − 2ac) / c²

Conclusion

Hence, if α and β are the zeros of the polynomial f(x) = ax² + bx + c, then the value of 1/α² + 1/β² is (b² − 2ac) / c².