3x^4-5x^2+1.5=0

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Solution for 3x^4-5x^2+1.5=0 equation:


Simplifying
3x4 + -5x2 + 1.5 = 0

Reorder the terms:
1.5 + -5x2 + 3x4 = 0

Solving
1.5 + -5x2 + 3x4 = 0

Solving for variable 'x'.

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

Divide each side by '3'.
0.5 + -1.666666667x2 + x4 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + -1.666666667x2 + -0.5 + x4 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + -1.666666667x2 + x4 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + -1.666666667x2 + x4 = 0 + -0.5
-1.666666667x2 + x4 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
-1.666666667x2 + x4 = -0.5

The x term is -1.666666667x2.  Take half its coefficient (-0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
-1.666666667x2 + 0.6944444447 + x4 = -0.5 + 0.6944444447

Reorder the terms:
0.6944444447 + -1.666666667x2 + x4 = -0.5 + 0.6944444447

Combine like terms: -0.5 + 0.6944444447 = 0.1944444447
0.6944444447 + -1.666666667x2 + x4 = 0.1944444447

Factor a perfect square on the left side:
(x2 + -0.8333333335)(x2 + -0.8333333335) = 0.1944444447

Calculate the square root of the right side: 0.440958552

Break this problem into two subproblems by setting 
(x2 + -0.8333333335) equal to 0.440958552 and -0.440958552.

Subproblem 1

x2 + -0.8333333335 = 0.440958552 Simplifying x2 + -0.8333333335 = 0.440958552 Reorder the terms: -0.8333333335 + x2 = 0.440958552 Solving -0.8333333335 + x2 = 0.440958552 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + x2 = 0.440958552 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + x2 = 0.440958552 + 0.8333333335 x2 = 0.440958552 + 0.8333333335 Combine like terms: 0.440958552 + 0.8333333335 = 1.2742918855 x2 = 1.2742918855 Simplifying x2 = 1.2742918855 Take the square root of each side: x = {-1.128845377, 1.128845377}

Subproblem 2

x2 + -0.8333333335 = -0.440958552 Simplifying x2 + -0.8333333335 = -0.440958552 Reorder the terms: -0.8333333335 + x2 = -0.440958552 Solving -0.8333333335 + x2 = -0.440958552 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + x2 = -0.440958552 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + x2 = -0.440958552 + 0.8333333335 x2 = -0.440958552 + 0.8333333335 Combine like terms: -0.440958552 + 0.8333333335 = 0.3923747815 x2 = 0.3923747815 Simplifying x2 = 0.3923747815 Take the square root of each side: x = {-0.626398261, 0.626398261}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-1.128845377, 1.128845377, -0.626398261, 0.626398261}

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