2x^4-30x^2+75=0

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Solution for 2x^4-30x^2+75=0 equation:


Simplifying
2x4 + -30x2 + 75 = 0

Reorder the terms:
75 + -30x2 + 2x4 = 0

Solving
75 + -30x2 + 2x4 = 0

Solving for variable 'x'.

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

Divide each side by '2'.
37.5 + -15x2 + x4 = 0

Move the constant term to the right:

Add '-37.5' to each side of the equation.
37.5 + -15x2 + -37.5 + x4 = 0 + -37.5

Reorder the terms:
37.5 + -37.5 + -15x2 + x4 = 0 + -37.5

Combine like terms: 37.5 + -37.5 = 0.0
0.0 + -15x2 + x4 = 0 + -37.5
-15x2 + x4 = 0 + -37.5

Combine like terms: 0 + -37.5 = -37.5
-15x2 + x4 = -37.5

The x term is -15x2.  Take half its coefficient (-7.5).
Square it (56.25) and add it to both sides.

Add '56.25' to each side of the equation.
-15x2 + 56.25 + x4 = -37.5 + 56.25

Reorder the terms:
56.25 + -15x2 + x4 = -37.5 + 56.25

Combine like terms: -37.5 + 56.25 = 18.75
56.25 + -15x2 + x4 = 18.75

Factor a perfect square on the left side:
(x2 + -7.5)(x2 + -7.5) = 18.75

Calculate the square root of the right side: 4.330127019

Break this problem into two subproblems by setting 
(x2 + -7.5) equal to 4.330127019 and -4.330127019.

Subproblem 1

x2 + -7.5 = 4.330127019 Simplifying x2 + -7.5 = 4.330127019 Reorder the terms: -7.5 + x2 = 4.330127019 Solving -7.5 + x2 = 4.330127019 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + x2 = 4.330127019 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + x2 = 4.330127019 + 7.5 x2 = 4.330127019 + 7.5 Combine like terms: 4.330127019 + 7.5 = 11.830127019 x2 = 11.830127019 Simplifying x2 = 11.830127019 Take the square root of each side: x = {-3.439495169, 3.439495169}

Subproblem 2

x2 + -7.5 = -4.330127019 Simplifying x2 + -7.5 = -4.330127019 Reorder the terms: -7.5 + x2 = -4.330127019 Solving -7.5 + x2 = -4.330127019 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + x2 = -4.330127019 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + x2 = -4.330127019 + 7.5 x2 = -4.330127019 + 7.5 Combine like terms: -4.330127019 + 7.5 = 3.169872981 x2 = 3.169872981 Simplifying x2 = 3.169872981 Take the square root of each side: x = {-1.780413711, 1.780413711}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-3.439495169, 3.439495169, -1.780413711, 1.780413711}

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