x^4-7x^2+5=0

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Solution for x^4-7x^2+5=0 equation:


Simplifying
x4 + -7x2 + 5 = 0

Reorder the terms:
5 + -7x2 + x4 = 0

Solving
5 + -7x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-5' to each side of the equation.
5 + -7x2 + -5 + x4 = 0 + -5

Reorder the terms:
5 + -5 + -7x2 + x4 = 0 + -5

Combine like terms: 5 + -5 = 0
0 + -7x2 + x4 = 0 + -5
-7x2 + x4 = 0 + -5

Combine like terms: 0 + -5 = -5
-7x2 + x4 = -5

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 = -5 + 12.25

Reorder the terms:
12.25 + -7x2 + x4 = -5 + 12.25

Combine like terms: -5 + 12.25 = 7.25
12.25 + -7x2 + x4 = 7.25

Factor a perfect square on the left side:
(x2 + -3.5)(x2 + -3.5) = 7.25

Calculate the square root of the right side: 2.692582404

Break this problem into two subproblems by setting 
(x2 + -3.5) equal to 2.692582404 and -2.692582404.

Subproblem 1

x2 + -3.5 = 2.692582404 Simplifying x2 + -3.5 = 2.692582404 Reorder the terms: -3.5 + x2 = 2.692582404 Solving -3.5 + x2 = 2.692582404 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 = 2.692582404 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x2 = 2.692582404 + 3.5 x2 = 2.692582404 + 3.5 Combine like terms: 2.692582404 + 3.5 = 6.192582404 x2 = 6.192582404 Simplifying x2 = 6.192582404 Take the square root of each side: x = {-2.488489985, 2.488489985}

Subproblem 2

x2 + -3.5 = -2.692582404 Simplifying x2 + -3.5 = -2.692582404 Reorder the terms: -3.5 + x2 = -2.692582404 Solving -3.5 + x2 = -2.692582404 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 = -2.692582404 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x2 = -2.692582404 + 3.5 x2 = -2.692582404 + 3.5 Combine like terms: -2.692582404 + 3.5 = 0.807417596 x2 = 0.807417596 Simplifying x2 = 0.807417596 Take the square root of each side: x = {-0.898564186, 0.898564186}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-2.488489985, 2.488489985, -0.898564186, 0.898564186}

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