Question

if alpha beta gamma are zeros of x3-12x2+44x+c,and the three zeros are in ap find c solve

Answer 1

alpha , beta and gamma are zeros of x³ - 12x² + 44x + c and the three zeros are in AP.
Let alpha = a - d , beta = a and gamma = a + d

now, sum of roots = alpha + beta + gamma = -(-12)/1 = 12
=> (a - d) + a + (a + d) = 12
=> 3a = 12
=> a = 4

sum of products of two consecutive roots = alpha.beta + beta.gamma +gamma.alpha
= 44
=> (a -d)a + a(a + d) + (a -d)(a + d) = 44
=> a² - ad + a² + ad + a² - d² = 44
=> 3a² - d² = 44
=> 3(4)² - d² = 44
=> - d² = 44 - 48 = -4
=> d = ±2

hence, a - d = 6 , a = 4, a + d = 2
so, alpha = 6, beta = 4 and gamma = 2
[note :- you can assume , gamma= 6,beta= 4 and alpha = 2 ]

now, product of roots = alpha.beta.gamma= c
=> 6 × 4 × 2 = c
=> c =48

hence, value of c = 48

Answer 2

HOPE IT HELPS YOU..

@RUKSHI