20x^4-145x^2+252=0

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Solution for 20x^4-145x^2+252=0 equation:


Simplifying
20x4 + -145x2 + 252 = 0

Reorder the terms:
252 + -145x2 + 20x4 = 0

Solving
252 + -145x2 + 20x4 = 0

Solving for variable 'x'.

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

Divide each side by '20'.
12.6 + -7.25x2 + x4 = 0

Move the constant term to the right:

Add '-12.6' to each side of the equation.
12.6 + -7.25x2 + -12.6 + x4 = 0 + -12.6

Reorder the terms:
12.6 + -12.6 + -7.25x2 + x4 = 0 + -12.6

Combine like terms: 12.6 + -12.6 = 0.0
0.0 + -7.25x2 + x4 = 0 + -12.6
-7.25x2 + x4 = 0 + -12.6

Combine like terms: 0 + -12.6 = -12.6
-7.25x2 + x4 = -12.6

The x term is -7.25x2.  Take half its coefficient (-3.625).
Square it (13.140625) and add it to both sides.

Add '13.140625' to each side of the equation.
-7.25x2 + 13.140625 + x4 = -12.6 + 13.140625

Reorder the terms:
13.140625 + -7.25x2 + x4 = -12.6 + 13.140625

Combine like terms: -12.6 + 13.140625 = 0.540625
13.140625 + -7.25x2 + x4 = 0.540625

Factor a perfect square on the left side:
(x2 + -3.625)(x2 + -3.625) = 0.540625

Calculate the square root of the right side: 0.735272058

Break this problem into two subproblems by setting 
(x2 + -3.625) equal to 0.735272058 and -0.735272058.

Subproblem 1

x2 + -3.625 = 0.735272058 Simplifying x2 + -3.625 = 0.735272058 Reorder the terms: -3.625 + x2 = 0.735272058 Solving -3.625 + x2 = 0.735272058 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.625' to each side of the equation. -3.625 + 3.625 + x2 = 0.735272058 + 3.625 Combine like terms: -3.625 + 3.625 = 0.000 0.000 + x2 = 0.735272058 + 3.625 x2 = 0.735272058 + 3.625 Combine like terms: 0.735272058 + 3.625 = 4.360272058 x2 = 4.360272058 Simplifying x2 = 4.360272058 Take the square root of each side: x = {-2.088126447, 2.088126447}

Subproblem 2

x2 + -3.625 = -0.735272058 Simplifying x2 + -3.625 = -0.735272058 Reorder the terms: -3.625 + x2 = -0.735272058 Solving -3.625 + x2 = -0.735272058 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.625' to each side of the equation. -3.625 + 3.625 + x2 = -0.735272058 + 3.625 Combine like terms: -3.625 + 3.625 = 0.000 0.000 + x2 = -0.735272058 + 3.625 x2 = -0.735272058 + 3.625 Combine like terms: -0.735272058 + 3.625 = 2.889727942 x2 = 2.889727942 Simplifying x2 = 2.889727942 Take the square root of each side: x = {-1.699919981, 1.699919981}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-2.088126447, 2.088126447, -1.699919981, 1.699919981}

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