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If the only positive real solution of ³√(x + 5) - ³√x = 1 is written in simplest (a+b√c)/d form, compute a + b + c + d.

If the only positive real solution of ³√(x + 5) - ³√x = 1 is written in simplest (a-example-1
User Uri
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1 Answer

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Given : ³√(x + 5) - ³√x = 1

We can write it in the simplest form as follows :

→(x+5) ^1/3 - (x)^1/3 -1 = 0

→ let u = ³√(x + 5) and x = u^3-5

Therefore:

u -³√u^3 -5

1. Then solve for u -³√u^3 -5=1

u = (3+ √57)/6 ...(1) and u = (3-√57)/6....(2)

2. Substitude u = ³√(x + 5) and solve :

(i) ³√(x + 5 = (3+ √57)/6

therefore x = -5/2 +(7√57)/18

(ii) ³√(x + 5 = (3- √57)/6

Therefore x = -5/2 +(7√57)/18

Therefore a = -5/2

b = 7

c =57

d = 18

User RoyHSIEH
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