Question

The product of the real roots of the equation (x +1)⁴ +(x+3)⁴ =
= 8 is
1) 0
2) 74
3) 7-2√3
4) 7+2√3​

Answer 1

Given : (x+1)⁴ +(x+3)⁴ = 8

To find : product of the real roots

Solution:

(x+1)⁴ +(x+3)⁴ = 8

let say

x = y - 2

=> y = x + 2

=> ( y - 1)⁴ + ( y + 1)⁴ = 8

=> 2y⁴ + 12y² + 2 = 8

=> y⁴ + 6y² + 1 = 4

=> y⁴ + 6y² - 3 = 0

y² = z

=> z² + 6z - 3 = 0

=> z = ( -6 ± √ (36 + 12) )/2

=> z = - 3 ± 2√3

z = - 3 - 2√3 => y is not real as y² is -ve

z = - 3 + 2√3

=> y² = 2√3 - 3

y = x + 2

=> (x + 2)² = 2√3 - 3

=> x² + 4x + 4 = 2√3 - 3

=> x² + 4x + 7 - 2√3 = 0

as D > 0 hence roots are real

Products of roots = 7 - 2√3

product of the real roots of the equation (x+1)⁴ +(x+3)⁴ = 8

is 7 - 2√3

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