Question

If a.ß are the roots of the equation x+px+p?+q = 0, then find the value of a^2+aß+ß^2+q​

Answer 1

Solution:

We have been given that α and β are the zeroes of polynomial x² + px + q.

Find relationship between Zeroes:

Sum of Zeros = -b/a

Sum of Zeros = -p/1

Sum of Zeros = -p

Product of Zeros = c/a

Product of Zeros = q/1

Product of Zeros = q

Now, We have to find the value of (α /ß + 2)(ß/α + 2)

(α /ß + 2)(ß/α + 2)

(α + 2ß)(ß + 2α)

a(ß + 2a) + 2ß(ß + 2a)

aß + 2a² + 2ß² + 4aß

(2a² + 2ß² )+ 5aß

2(a²+ß²) + 5aß

2[(a+ß)² - 2ab ] + 5ab

2[(-p)² - 2(q) ] + 5(q)

2[(p²) - 2q ] + 5q

2[p² - 2q] + 5q

2p² - 4q + 5q

2p² + q

Therefore, Required Value of (α /ß + 2)(ß/α + 2) is 2p² + q.