x^4+3=8x^2

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Solution for x^4+3=8x^2 equation:


Simplifying
x4 + 3 = 8x2

Reorder the terms:
3 + x4 = 8x2

Solving
3 + x4 = 8x2

Solving for variable 'x'.

Reorder the terms:
3 + -8x2 + x4 = 8x2 + -8x2

Combine like terms: 8x2 + -8x2 = 0
3 + -8x2 + x4 = 0

Begin completing the square.

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + -8x2 + -3 + x4 = 0 + -3

Reorder the terms:
3 + -3 + -8x2 + x4 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + -8x2 + x4 = 0 + -3
-8x2 + x4 = 0 + -3

Combine like terms: 0 + -3 = -3
-8x2 + x4 = -3

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 = -3 + 16

Reorder the terms:
16 + -8x2 + x4 = -3 + 16

Combine like terms: -3 + 16 = 13
16 + -8x2 + x4 = 13

Factor a perfect square on the left side:
(x2 + -4)(x2 + -4) = 13

Calculate the square root of the right side: 3.605551275

Break this problem into two subproblems by setting 
(x2 + -4) equal to 3.605551275 and -3.605551275.

Subproblem 1

x2 + -4 = 3.605551275 Simplifying x2 + -4 = 3.605551275 Reorder the terms: -4 + x2 = 3.605551275 Solving -4 + x2 = 3.605551275 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 = 3.605551275 + 4 Combine like terms: -4 + 4 = 0 0 + x2 = 3.605551275 + 4 x2 = 3.605551275 + 4 Combine like terms: 3.605551275 + 4 = 7.605551275 x2 = 7.605551275 Simplifying x2 = 7.605551275 Take the square root of each side: x = {-2.757816396, 2.757816396}

Subproblem 2

x2 + -4 = -3.605551275 Simplifying x2 + -4 = -3.605551275 Reorder the terms: -4 + x2 = -3.605551275 Solving -4 + x2 = -3.605551275 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 = -3.605551275 + 4 Combine like terms: -4 + 4 = 0 0 + x2 = -3.605551275 + 4 x2 = -3.605551275 + 4 Combine like terms: -3.605551275 + 4 = 0.394448725 x2 = 0.394448725 Simplifying x2 = 0.394448725 Take the square root of each side: x = {-0.628051531, 0.628051531}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-2.757816396, 2.757816396, -0.628051531, 0.628051531}

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