x^4-6.25x^2+9=0

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Solution for x^4-6.25x^2+9=0 equation:


Simplifying
x4 + -6.25x2 + 9 = 0

Reorder the terms:
9 + -6.25x2 + x4 = 0

Solving
9 + -6.25x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-9' to each side of the equation.
9 + -6.25x2 + -9 + x4 = 0 + -9

Reorder the terms:
9 + -9 + -6.25x2 + x4 = 0 + -9

Combine like terms: 9 + -9 = 0
0 + -6.25x2 + x4 = 0 + -9
-6.25x2 + x4 = 0 + -9

Combine like terms: 0 + -9 = -9
-6.25x2 + x4 = -9

The x term is -6.25x2.  Take half its coefficient (-3.125).
Square it (9.765625) and add it to both sides.

Add '9.765625' to each side of the equation.
-6.25x2 + 9.765625 + x4 = -9 + 9.765625

Reorder the terms:
9.765625 + -6.25x2 + x4 = -9 + 9.765625

Combine like terms: -9 + 9.765625 = 0.765625
9.765625 + -6.25x2 + x4 = 0.765625

Factor a perfect square on the left side:
(x2 + -3.125)(x2 + -3.125) = 0.765625

Calculate the square root of the right side: 0.875

Break this problem into two subproblems by setting 
(x2 + -3.125) equal to 0.875 and -0.875.

Subproblem 1

x2 + -3.125 = 0.875 Simplifying x2 + -3.125 = 0.875 Reorder the terms: -3.125 + x2 = 0.875 Solving -3.125 + x2 = 0.875 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.125' to each side of the equation. -3.125 + 3.125 + x2 = 0.875 + 3.125 Combine like terms: -3.125 + 3.125 = 0.000 0.000 + x2 = 0.875 + 3.125 x2 = 0.875 + 3.125 Combine like terms: 0.875 + 3.125 = 4 x2 = 4 Simplifying x2 = 4 Take the square root of each side: x = {-2, 2}

Subproblem 2

x2 + -3.125 = -0.875 Simplifying x2 + -3.125 = -0.875 Reorder the terms: -3.125 + x2 = -0.875 Solving -3.125 + x2 = -0.875 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.125' to each side of the equation. -3.125 + 3.125 + x2 = -0.875 + 3.125 Combine like terms: -3.125 + 3.125 = 0.000 0.000 + x2 = -0.875 + 3.125 x2 = -0.875 + 3.125 Combine like terms: -0.875 + 3.125 = 2.25 x2 = 2.25 Simplifying x2 = 2.25 Take the square root of each side: x = {-1.5, 1.5}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-2, 2, -1.5, 1.5}

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