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

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


Simplifying
x4 + -7x2 + 4 = 0

Reorder the terms:
4 + -7x2 + x4 = 0

Solving
4 + -7x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-4' to each side of the equation.
4 + -7x2 + -4 + x4 = 0 + -4

Reorder the terms:
4 + -4 + -7x2 + x4 = 0 + -4

Combine like terms: 4 + -4 = 0
0 + -7x2 + x4 = 0 + -4
-7x2 + x4 = 0 + -4

Combine like terms: 0 + -4 = -4
-7x2 + x4 = -4

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 = -4 + 12.25

Reorder the terms:
12.25 + -7x2 + x4 = -4 + 12.25

Combine like terms: -4 + 12.25 = 8.25
12.25 + -7x2 + x4 = 8.25

Factor a perfect square on the left side:
(x2 + -3.5)(x2 + -3.5) = 8.25

Calculate the square root of the right side: 2.872281323

Break this problem into two subproblems by setting 
(x2 + -3.5) equal to 2.872281323 and -2.872281323.

Subproblem 1

x2 + -3.5 = 2.872281323 Simplifying x2 + -3.5 = 2.872281323 Reorder the terms: -3.5 + x2 = 2.872281323 Solving -3.5 + x2 = 2.872281323 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 = 2.872281323 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x2 = 2.872281323 + 3.5 x2 = 2.872281323 + 3.5 Combine like terms: 2.872281323 + 3.5 = 6.372281323 x2 = 6.372281323 Simplifying x2 = 6.372281323 Take the square root of each side: x = {-2.524337799, 2.524337799}

Subproblem 2

x2 + -3.5 = -2.872281323 Simplifying x2 + -3.5 = -2.872281323 Reorder the terms: -3.5 + x2 = -2.872281323 Solving -3.5 + x2 = -2.872281323 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 = -2.872281323 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x2 = -2.872281323 + 3.5 x2 = -2.872281323 + 3.5 Combine like terms: -2.872281323 + 3.5 = 0.627718677 x2 = 0.627718677 Simplifying x2 = 0.627718677 Take the square root of each side: x = {-0.792286992, 0.792286992}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-2.524337799, 2.524337799, -0.792286992, 0.792286992}

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