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

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


Simplifying
3y4 + -5y2 + 1.5 = 0

Reorder the terms:
1.5 + -5y2 + 3y4 = 0

Solving
1.5 + -5y2 + 3y4 = 0

Solving for variable 'y'.

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

Divide each side by '3'.
0.5 + -1.666666667y2 + y4 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + -1.666666667y2 + -0.5 + y4 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + -1.666666667y2 + y4 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + -1.666666667y2 + y4 = 0 + -0.5
-1.666666667y2 + y4 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
-1.666666667y2 + y4 = -0.5

The y term is -1.666666667y2.  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.
-1.666666667y2 + 0.6944444447 + y4 = -0.5 + 0.6944444447

Reorder the terms:
0.6944444447 + -1.666666667y2 + y4 = -0.5 + 0.6944444447

Combine like terms: -0.5 + 0.6944444447 = 0.1944444447
0.6944444447 + -1.666666667y2 + y4 = 0.1944444447

Factor a perfect square on the left side:
(y2 + -0.8333333335)(y2 + -0.8333333335) = 0.1944444447

Calculate the square root of the right side: 0.440958552

Break this problem into two subproblems by setting 
(y2 + -0.8333333335) equal to 0.440958552 and -0.440958552.

Subproblem 1

y2 + -0.8333333335 = 0.440958552 Simplifying y2 + -0.8333333335 = 0.440958552 Reorder the terms: -0.8333333335 + y2 = 0.440958552 Solving -0.8333333335 + y2 = 0.440958552 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + y2 = 0.440958552 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + y2 = 0.440958552 + 0.8333333335 y2 = 0.440958552 + 0.8333333335 Combine like terms: 0.440958552 + 0.8333333335 = 1.2742918855 y2 = 1.2742918855 Simplifying y2 = 1.2742918855 Take the square root of each side: y = {-1.128845377, 1.128845377}

Subproblem 2

y2 + -0.8333333335 = -0.440958552 Simplifying y2 + -0.8333333335 = -0.440958552 Reorder the terms: -0.8333333335 + y2 = -0.440958552 Solving -0.8333333335 + y2 = -0.440958552 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + y2 = -0.440958552 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + y2 = -0.440958552 + 0.8333333335 y2 = -0.440958552 + 0.8333333335 Combine like terms: -0.440958552 + 0.8333333335 = 0.3923747815 y2 = 0.3923747815 Simplifying y2 = 0.3923747815 Take the square root of each side: y = {-0.626398261, 0.626398261}

Solution

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

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