Question

solve this please
it's little urgent!

Answer 1

Answer:

(p⁴/q²) - (4p²/q) + 2

Step-by-step explanation:

Given Equation is x² - px + Q.

Here, a = 1, b = -p, c = q.

Given that α,β are the roots of the equation.

(i)

Sum of roots = -b/a

α + β = -(-p)/1

α + β = p.

(ii)

Product of roots = c/a

αβ = q/1

αβ = q.

LHS:

(α²/β²) + (β²/α²)

= [(α² + β²)² - 2α²β²]/α²β²

= {[(α+β)² - 2αβ]² - 2(αβ)}²}/(αβ)²

= {[(p)² -2(q)]²-2(q)²}/(q)²

= [(p² - 2q)² - 2(q)²]/q²

= [(p⁴ + 4q² - 4p²q - 2q²)]/q²

= [(p⁴ - 4p²q + 2q²)]/q²

= (p⁴/q²) - (4p²q/q²) + (2q²/q²)

= (p⁴/q²) - (4p²/q) + 2.

Hope it helps!

Answer 2

alpha + beta = p. ........(1)

alpha beta = Q........(2)

Now divide 2 by 1

( alpha + beta)/ alpha beta = P/Q

SQUARING BOTH SIDE

alpha ^2 + beta ^2 + 2 alpha beta )/ alpha^2 beta^2 = (P^2)/Q^2

1/beta^2 + 1/alpha^2+ 2 /alpha beta = = (p/Q)^2

alpha^2 + beta^2)/ alpha^2 beta^2 = ( p/q)^2 - 2 /Q

alpha^2 + beta^2 = ( p/q)^2 -2/Q ) × Q^2
We have to find alpha^4 + beta^4)/ alpha^2 beta^2

= ( alpha^2+ beta^2)^2 - 2alpha^2beta^2)/ apha^2 beta^2

= ( P/q)^2 -2/Q) Q^2 )^2- 2 Q^2 / Q^2

= ( P^2/Q^2- 2/Q)^2 Q^4 - 2Q^2)/ Q^2

= P^2 /Q^2 - 2/Q)^2 Q^2 - 2

= P^4/Q^4 + 4/Q^2 - 4P^2/Q ) Q^2 - 2

= P^4/Q^2 + 4 - 4p^2 Q - 2

= P^4/Q^2 - 4P^2/Q + 2

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