x^4=8x^2-6

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Solution for x^4=8x^2-6 equation:


Simplifying
x4 = 8x2 + -6

Reorder the terms:
x4 = -6 + 8x2

Solving
x4 = -6 + 8x2

Solving for variable 'x'.

Reorder the terms:
6 + -8x2 + x4 = -6 + 8x2 + 6 + -8x2

Reorder the terms:
6 + -8x2 + x4 = -6 + 6 + 8x2 + -8x2

Combine like terms: -6 + 6 = 0
6 + -8x2 + x4 = 0 + 8x2 + -8x2
6 + -8x2 + x4 = 8x2 + -8x2

Combine like terms: 8x2 + -8x2 = 0
6 + -8x2 + x4 = 0

Begin completing the square.

Move the constant term to the right:

Add '-6' to each side of the equation.
6 + -8x2 + -6 + x4 = 0 + -6

Reorder the terms:
6 + -6 + -8x2 + x4 = 0 + -6

Combine like terms: 6 + -6 = 0
0 + -8x2 + x4 = 0 + -6
-8x2 + x4 = 0 + -6

Combine like terms: 0 + -6 = -6
-8x2 + x4 = -6

The x term is -8x2.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8x2 + 16 + x4 = -6 + 16

Reorder the terms:
16 + -8x2 + x4 = -6 + 16

Combine like terms: -6 + 16 = 10
16 + -8x2 + x4 = 10

Factor a perfect square on the left side:
(x2 + -4)(x2 + -4) = 10

Calculate the square root of the right side: 3.16227766

Break this problem into two subproblems by setting 
(x2 + -4) equal to 3.16227766 and -3.16227766.

Subproblem 1

x2 + -4 = 3.16227766 Simplifying x2 + -4 = 3.16227766 Reorder the terms: -4 + x2 = 3.16227766 Solving -4 + x2 = 3.16227766 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + x2 = 3.16227766 + 4 Combine like terms: -4 + 4 = 0 0 + x2 = 3.16227766 + 4 x2 = 3.16227766 + 4 Combine like terms: 3.16227766 + 4 = 7.16227766 x2 = 7.16227766 Simplifying x2 = 7.16227766 Take the square root of each side: x = {-2.676243199, 2.676243199}

Subproblem 2

x2 + -4 = -3.16227766 Simplifying x2 + -4 = -3.16227766 Reorder the terms: -4 + x2 = -3.16227766 Solving -4 + x2 = -3.16227766 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + x2 = -3.16227766 + 4 Combine like terms: -4 + 4 = 0 0 + x2 = -3.16227766 + 4 x2 = -3.16227766 + 4 Combine like terms: -3.16227766 + 4 = 0.83772234 x2 = 0.83772234 Simplifying x2 = 0.83772234 Take the square root of each side: x = {-0.91527173, 0.91527173}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-2.676243199, 2.676243199, -0.91527173, 0.91527173}

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