x^4-7x^2+7=0

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Solution for x^4-7x^2+7=0 equation:


Simplifying
x4 + -7x2 + 7 = 0

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

Solving
7 + -7x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

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

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

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

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

The x term is -7x2.  Take half its coefficient (-3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
-7x2 + 12.25 + x4 = -7 + 12.25

Reorder the terms:
12.25 + -7x2 + x4 = -7 + 12.25

Combine like terms: -7 + 12.25 = 5.25
12.25 + -7x2 + x4 = 5.25

Factor a perfect square on the left side:
(x2 + -3.5)(x2 + -3.5) = 5.25

Calculate the square root of the right side: 2.291287847

Break this problem into two subproblems by setting 
(x2 + -3.5) equal to 2.291287847 and -2.291287847.

Subproblem 1

x2 + -3.5 = 2.291287847 Simplifying x2 + -3.5 = 2.291287847 Reorder the terms: -3.5 + x2 = 2.291287847 Solving -3.5 + x2 = 2.291287847 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + x2 = 2.291287847 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x2 = 2.291287847 + 3.5 x2 = 2.291287847 + 3.5 Combine like terms: 2.291287847 + 3.5 = 5.791287847 x2 = 5.791287847 Simplifying x2 = 5.791287847 Take the square root of each side: x = {-2.406509474, 2.406509474}

Subproblem 2

x2 + -3.5 = -2.291287847 Simplifying x2 + -3.5 = -2.291287847 Reorder the terms: -3.5 + x2 = -2.291287847 Solving -3.5 + x2 = -2.291287847 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + x2 = -2.291287847 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x2 = -2.291287847 + 3.5 x2 = -2.291287847 + 3.5 Combine like terms: -2.291287847 + 3.5 = 1.208712153 x2 = 1.208712153 Simplifying x2 = 1.208712153 Take the square root of each side: x = {-1.099414459, 1.099414459}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-2.406509474, 2.406509474, -1.099414459, 1.099414459}

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