x8+x4+1=0

This solution đơn hàng with Equations reducible to quadratic form.


Step by Step Solution

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Reformatting the input đầu vào :

Changes made lớn your đầu vào should not affect the solution: (1): "x4" was replaced by "x^4". 1 more similar replacement(s).

Step by step solution :

Step 1 :

Trying to lớn factor by splitting the middle term

1.1Factoring x8+x4+1 The first term is, x8 its coefficient is 1.The middle term is, +x4 its coefficient is 1.The last term, "the constant", is +1Step-1 : Multiply the coefficient of the first term by the constant 1•1=1Step-2 : Find two factors of 1 whose sum equals the coefficient of the middle term, which is 1.

-1+-1=-2
1+1=2

Observation : No two such factors can be found !! Conclusion : Trinomial can not be factored

Equation at the end of step 1 :

x8 + x4 + 1 = 0

Step 2 :

Solving a Single Variable Equation:

Equations which are reducible lớn quadratic :

2.1Solvex8+x4+1 = 0This equation is reducible to lớn quadratic. What this means is that using a new variable, we can rewrite this equation as a quadratic equation Using w, such that w = x4 transforms the equation into:w2+w+1 = 0Solving this new equation using the quadratic formula we get two imaginary solutions:w=-0.5000± 0.8660 iNow that we know the value(s) of w, we can calculate x since x is ∜ w Since we are speaking 4th root, each of the two imaginary solutions of has 4 rootsTiger finds these roots using de Moivre"s FormulaThe 4th roots of -0.500 + 0.866 i are:

x = 0.866 + 0.500 i x = -0.500 + 0.866 i x = -0.866 -0.500 i x = 0.500 -0.866 i 4th roots of -0.500- 0.866 i : x = 0.500 + 0.866 i x = -0.866 + 0.500 i x = -0.500 - 0.866 i x = 0.866 - 0.500 i

8 solutions were found :

x = 0.866 - 0.500 i x = -0.500 - 0.866 i x = -0.866 + 0.500 i x = 0.500 + 0.866 i x = 0.500 -0.866 i x = -0.866 -0.500 i x = -0.500 + 0.866 i x = 0.866 + 0.500 i