x^4-20x^2+10=0

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Solution for x^4-20x^2+10=0 equation:


Simplifying
x4 + -20x2 + 10 = 0

Reorder the terms:
10 + -20x2 + x4 = 0

Solving
10 + -20x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-10' to each side of the equation.
10 + -20x2 + -10 + x4 = 0 + -10

Reorder the terms:
10 + -10 + -20x2 + x4 = 0 + -10

Combine like terms: 10 + -10 = 0
0 + -20x2 + x4 = 0 + -10
-20x2 + x4 = 0 + -10

Combine like terms: 0 + -10 = -10
-20x2 + x4 = -10

The x term is -20x2.  Take half its coefficient (-10).
Square it (100) and add it to both sides.

Add '100' to each side of the equation.
-20x2 + 100 + x4 = -10 + 100

Reorder the terms:
100 + -20x2 + x4 = -10 + 100

Combine like terms: -10 + 100 = 90
100 + -20x2 + x4 = 90

Factor a perfect square on the left side:
(x2 + -10)(x2 + -10) = 90

Calculate the square root of the right side: 9.486832981

Break this problem into two subproblems by setting 
(x2 + -10) equal to 9.486832981 and -9.486832981.

Subproblem 1

x2 + -10 = 9.486832981 Simplifying x2 + -10 = 9.486832981 Reorder the terms: -10 + x2 = 9.486832981 Solving -10 + x2 = 9.486832981 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '10' to each side of the equation. -10 + 10 + x2 = 9.486832981 + 10 Combine like terms: -10 + 10 = 0 0 + x2 = 9.486832981 + 10 x2 = 9.486832981 + 10 Combine like terms: 9.486832981 + 10 = 19.486832981 x2 = 19.486832981 Simplifying x2 = 19.486832981 Take the square root of each side: x = {-4.41438931, 4.41438931}

Subproblem 2

x2 + -10 = -9.486832981 Simplifying x2 + -10 = -9.486832981 Reorder the terms: -10 + x2 = -9.486832981 Solving -10 + x2 = -9.486832981 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '10' to each side of the equation. -10 + 10 + x2 = -9.486832981 + 10 Combine like terms: -10 + 10 = 0 0 + x2 = -9.486832981 + 10 x2 = -9.486832981 + 10 Combine like terms: -9.486832981 + 10 = 0.513167019 x2 = 0.513167019 Simplifying x2 = 0.513167019 Take the square root of each side: x = {-0.716356768, 0.716356768}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-4.41438931, 4.41438931, -0.716356768, 0.716356768}

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