If α and β are the zeros of the quadratic polynomial f(x) = x² − px + q, prove that α²/β² + β²/α² = p⁴/q² − 4p²/q + 2

Video Explanation

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Proof

Given polynomial:

f(x) = x² − px + q

Let α and β be the zeros of the given polynomial.

Step 1: Find α + β and αβ

Comparing f(x) = x² − px + q with ax² + bx + c:

a = 1,   b = −p,   c = q

α + β = −b/a = p

αβ = c/a = q

Step 2: Find α²/β² + β²/α²

α²/β² + β²/α²

= (α⁴ + β⁴)/(α²β²)

= (α⁴ + β⁴)/(αβ)²

Since,

α⁴ + β⁴ = (α² + β²)² − 2α²β²

and

α² + β² = (α + β)² − 2αβ

Step 3: Substitute the Values

α² + β² = p² − 2q

α⁴ + β⁴ = (p² − 2q)² − 2q²

= p⁴ − 4p²q + 4q² − 2q²

= p⁴ − 4p²q + 2q²

Also,

(αβ)² = q²

Step 4: Final Calculation

α²/β² + β²/α²

= (p⁴ − 4p²q + 2q²)/q²

= p⁴/q² − 4p²/q + 2

Hence Proved

α²/β² + β²/α² = p⁴/q² − 4p²/q + 2

Conclusion

Thus, using the relationship between zeros and coefficients of the quadratic polynomial, the given identity is proved.