=x^4-45x^2+100

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Solution for =x^4-45x^2+100 equation:


Simplifying
0 = x4 + -45x2 + 100

Reorder the terms:
0 = 100 + -45x2 + x4

Solving
0 = 100 + -45x2 + x4

Solving for variable 'x'.

Combine like terms: 0 + -100 = -100
-100 + 45x2 + -1x4 = 100 + -45x2 + x4 + -100 + 45x2 + -1x4

Reorder the terms:
-100 + 45x2 + -1x4 = 100 + -100 + -45x2 + 45x2 + x4 + -1x4

Combine like terms: 100 + -100 = 0
-100 + 45x2 + -1x4 = 0 + -45x2 + 45x2 + x4 + -1x4
-100 + 45x2 + -1x4 = -45x2 + 45x2 + x4 + -1x4

Combine like terms: -45x2 + 45x2 = 0
-100 + 45x2 + -1x4 = 0 + x4 + -1x4
-100 + 45x2 + -1x4 = x4 + -1x4

Combine like terms: x4 + -1x4 = 0
-100 + 45x2 + -1x4 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
100 + -45x2 + x4 = 0

Move the constant term to the right:

Add '-100' to each side of the equation.
100 + -45x2 + -100 + x4 = 0 + -100

Reorder the terms:
100 + -100 + -45x2 + x4 = 0 + -100

Combine like terms: 100 + -100 = 0
0 + -45x2 + x4 = 0 + -100
-45x2 + x4 = 0 + -100

Combine like terms: 0 + -100 = -100
-45x2 + x4 = -100

The x term is -45x2.  Take half its coefficient (-22.5).
Square it (506.25) and add it to both sides.

Add '506.25' to each side of the equation.
-45x2 + 506.25 + x4 = -100 + 506.25

Reorder the terms:
506.25 + -45x2 + x4 = -100 + 506.25

Combine like terms: -100 + 506.25 = 406.25
506.25 + -45x2 + x4 = 406.25

Factor a perfect square on the left side:
(x2 + -22.5)(x2 + -22.5) = 406.25

Calculate the square root of the right side: 20.155644371

Break this problem into two subproblems by setting 
(x2 + -22.5) equal to 20.155644371 and -20.155644371.

Subproblem 1

x2 + -22.5 = 20.155644371 Simplifying x2 + -22.5 = 20.155644371 Reorder the terms: -22.5 + x2 = 20.155644371 Solving -22.5 + x2 = 20.155644371 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '22.5' to each side of the equation. -22.5 + 22.5 + x2 = 20.155644371 + 22.5 Combine like terms: -22.5 + 22.5 = 0.0 0.0 + x2 = 20.155644371 + 22.5 x2 = 20.155644371 + 22.5 Combine like terms: 20.155644371 + 22.5 = 42.655644371 x2 = 42.655644371 Simplifying x2 = 42.655644371 Take the square root of each side: x = {-6.531128874, 6.531128874}

Subproblem 2

x2 + -22.5 = -20.155644371 Simplifying x2 + -22.5 = -20.155644371 Reorder the terms: -22.5 + x2 = -20.155644371 Solving -22.5 + x2 = -20.155644371 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '22.5' to each side of the equation. -22.5 + 22.5 + x2 = -20.155644371 + 22.5 Combine like terms: -22.5 + 22.5 = 0.0 0.0 + x2 = -20.155644371 + 22.5 x2 = -20.155644371 + 22.5 Combine like terms: -20.155644371 + 22.5 = 2.344355629 x2 = 2.344355629 Simplifying x2 = 2.344355629 Take the square root of each side: x = {-1.531128874, 1.531128874}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-6.531128874, 6.531128874, -1.531128874, 1.531128874}

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