4x^4-14x^2+5=0

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Solution for 4x^4-14x^2+5=0 equation:


Simplifying
4x4 + -14x2 + 5 = 0

Reorder the terms:
5 + -14x2 + 4x4 = 0

Solving
5 + -14x2 + 4x4 = 0

Solving for variable 'x'.

Begin completing the square.  Divide all terms by
4 the coefficient of the squared term: 

Divide each side by '4'.
1.25 + -3.5x2 + x4 = 0

Move the constant term to the right:

Add '-1.25' to each side of the equation.
1.25 + -3.5x2 + -1.25 + x4 = 0 + -1.25

Reorder the terms:
1.25 + -1.25 + -3.5x2 + x4 = 0 + -1.25

Combine like terms: 1.25 + -1.25 = 0.00
0.00 + -3.5x2 + x4 = 0 + -1.25
-3.5x2 + x4 = 0 + -1.25

Combine like terms: 0 + -1.25 = -1.25
-3.5x2 + x4 = -1.25

The x term is -3.5x2.  Take half its coefficient (-1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
-3.5x2 + 3.0625 + x4 = -1.25 + 3.0625

Reorder the terms:
3.0625 + -3.5x2 + x4 = -1.25 + 3.0625

Combine like terms: -1.25 + 3.0625 = 1.8125
3.0625 + -3.5x2 + x4 = 1.8125

Factor a perfect square on the left side:
(x2 + -1.75)(x2 + -1.75) = 1.8125

Calculate the square root of the right side: 1.346291202

Break this problem into two subproblems by setting 
(x2 + -1.75) equal to 1.346291202 and -1.346291202.

Subproblem 1

x2 + -1.75 = 1.346291202 Simplifying x2 + -1.75 = 1.346291202 Reorder the terms: -1.75 + x2 = 1.346291202 Solving -1.75 + x2 = 1.346291202 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + x2 = 1.346291202 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + x2 = 1.346291202 + 1.75 x2 = 1.346291202 + 1.75 Combine like terms: 1.346291202 + 1.75 = 3.096291202 x2 = 3.096291202 Simplifying x2 = 3.096291202 Take the square root of each side: x = {-1.759628143, 1.759628143}

Subproblem 2

x2 + -1.75 = -1.346291202 Simplifying x2 + -1.75 = -1.346291202 Reorder the terms: -1.75 + x2 = -1.346291202 Solving -1.75 + x2 = -1.346291202 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + x2 = -1.346291202 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + x2 = -1.346291202 + 1.75 x2 = -1.346291202 + 1.75 Combine like terms: -1.346291202 + 1.75 = 0.403708798 x2 = 0.403708798 Simplifying x2 = 0.403708798 Take the square root of each side: x = {-0.635380829, 0.635380829}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-1.759628143, 1.759628143, -0.635380829, 0.635380829}

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