x^4=6x^2-7

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Solution for x^4=6x^2-7 equation:


Simplifying
x4 = 6x2 + -7

Reorder the terms:
x4 = -7 + 6x2

Solving
x4 = -7 + 6x2

Solving for variable 'x'.

Reorder the terms:
7 + -6x2 + x4 = -7 + 6x2 + 7 + -6x2

Reorder the terms:
7 + -6x2 + x4 = -7 + 7 + 6x2 + -6x2

Combine like terms: -7 + 7 = 0
7 + -6x2 + x4 = 0 + 6x2 + -6x2
7 + -6x2 + x4 = 6x2 + -6x2

Combine like terms: 6x2 + -6x2 = 0
7 + -6x2 + x4 = 0

Begin completing the square.

Move the constant term to the right:

Add '-7' to each side of the equation.
7 + -6x2 + -7 + x4 = 0 + -7

Reorder the terms:
7 + -7 + -6x2 + x4 = 0 + -7

Combine like terms: 7 + -7 = 0
0 + -6x2 + x4 = 0 + -7
-6x2 + x4 = 0 + -7

Combine like terms: 0 + -7 = -7
-6x2 + x4 = -7

The x term is -6x2.  Take half its coefficient (-3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
-6x2 + 9 + x4 = -7 + 9

Reorder the terms:
9 + -6x2 + x4 = -7 + 9

Combine like terms: -7 + 9 = 2
9 + -6x2 + x4 = 2

Factor a perfect square on the left side:
(x2 + -3)(x2 + -3) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(x2 + -3) equal to 1.414213562 and -1.414213562.

Subproblem 1

x2 + -3 = 1.414213562 Simplifying x2 + -3 = 1.414213562 Reorder the terms: -3 + x2 = 1.414213562 Solving -3 + x2 = 1.414213562 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + x2 = 1.414213562 + 3 Combine like terms: -3 + 3 = 0 0 + x2 = 1.414213562 + 3 x2 = 1.414213562 + 3 Combine like terms: 1.414213562 + 3 = 4.414213562 x2 = 4.414213562 Simplifying x2 = 4.414213562 Take the square root of each side: x = {-2.10100299, 2.10100299}

Subproblem 2

x2 + -3 = -1.414213562 Simplifying x2 + -3 = -1.414213562 Reorder the terms: -3 + x2 = -1.414213562 Solving -3 + x2 = -1.414213562 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + x2 = -1.414213562 + 3 Combine like terms: -3 + 3 = 0 0 + x2 = -1.414213562 + 3 x2 = -1.414213562 + 3 Combine like terms: -1.414213562 + 3 = 1.585786438 x2 = 1.585786438 Simplifying x2 = 1.585786438 Take the square root of each side: x = {-1.259280127, 1.259280127}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-2.10100299, 2.10100299, -1.259280127, 1.259280127}

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