10x^4-33x^2+15=0

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Solution for 10x^4-33x^2+15=0 equation:


Simplifying
10x4 + -33x2 + 15 = 0

Reorder the terms:
15 + -33x2 + 10x4 = 0

Solving
15 + -33x2 + 10x4 = 0

Solving for variable 'x'.

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

Divide each side by '10'.
1.5 + -3.3x2 + x4 = 0

Move the constant term to the right:

Add '-1.5' to each side of the equation.
1.5 + -3.3x2 + -1.5 + x4 = 0 + -1.5

Reorder the terms:
1.5 + -1.5 + -3.3x2 + x4 = 0 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + -3.3x2 + x4 = 0 + -1.5
-3.3x2 + x4 = 0 + -1.5

Combine like terms: 0 + -1.5 = -1.5
-3.3x2 + x4 = -1.5

The x term is -3.3x2.  Take half its coefficient (-1.65).
Square it (2.7225) and add it to both sides.

Add '2.7225' to each side of the equation.
-3.3x2 + 2.7225 + x4 = -1.5 + 2.7225

Reorder the terms:
2.7225 + -3.3x2 + x4 = -1.5 + 2.7225

Combine like terms: -1.5 + 2.7225 = 1.2225
2.7225 + -3.3x2 + x4 = 1.2225

Factor a perfect square on the left side:
(x2 + -1.65)(x2 + -1.65) = 1.2225

Calculate the square root of the right side: 1.105667219

Break this problem into two subproblems by setting 
(x2 + -1.65) equal to 1.105667219 and -1.105667219.

Subproblem 1

x2 + -1.65 = 1.105667219 Simplifying x2 + -1.65 = 1.105667219 Reorder the terms: -1.65 + x2 = 1.105667219 Solving -1.65 + x2 = 1.105667219 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.65' to each side of the equation. -1.65 + 1.65 + x2 = 1.105667219 + 1.65 Combine like terms: -1.65 + 1.65 = 0.00 0.00 + x2 = 1.105667219 + 1.65 x2 = 1.105667219 + 1.65 Combine like terms: 1.105667219 + 1.65 = 2.755667219 x2 = 2.755667219 Simplifying x2 = 2.755667219 Take the square root of each side: x = {-1.660020247, 1.660020247}

Subproblem 2

x2 + -1.65 = -1.105667219 Simplifying x2 + -1.65 = -1.105667219 Reorder the terms: -1.65 + x2 = -1.105667219 Solving -1.65 + x2 = -1.105667219 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.65' to each side of the equation. -1.65 + 1.65 + x2 = -1.105667219 + 1.65 Combine like terms: -1.65 + 1.65 = 0.00 0.00 + x2 = -1.105667219 + 1.65 x2 = -1.105667219 + 1.65 Combine like terms: -1.105667219 + 1.65 = 0.544332781 x2 = 0.544332781 Simplifying x2 = 0.544332781 Take the square root of each side: x = {-0.737789117, 0.737789117}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-1.660020247, 1.660020247, -0.737789117, 0.737789117}

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