x^4-15x^2+24=0

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Solution for x^4-15x^2+24=0 equation:


Simplifying
x4 + -15x2 + 24 = 0

Reorder the terms:
24 + -15x2 + x4 = 0

Solving
24 + -15x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-24' to each side of the equation.
24 + -15x2 + -24 + x4 = 0 + -24

Reorder the terms:
24 + -24 + -15x2 + x4 = 0 + -24

Combine like terms: 24 + -24 = 0
0 + -15x2 + x4 = 0 + -24
-15x2 + x4 = 0 + -24

Combine like terms: 0 + -24 = -24
-15x2 + x4 = -24

The x term is -15x2.  Take half its coefficient (-7.5).
Square it (56.25) and add it to both sides.

Add '56.25' to each side of the equation.
-15x2 + 56.25 + x4 = -24 + 56.25

Reorder the terms:
56.25 + -15x2 + x4 = -24 + 56.25

Combine like terms: -24 + 56.25 = 32.25
56.25 + -15x2 + x4 = 32.25

Factor a perfect square on the left side:
(x2 + -7.5)(x2 + -7.5) = 32.25

Calculate the square root of the right side: 5.678908346

Break this problem into two subproblems by setting 
(x2 + -7.5) equal to 5.678908346 and -5.678908346.

Subproblem 1

x2 + -7.5 = 5.678908346 Simplifying x2 + -7.5 = 5.678908346 Reorder the terms: -7.5 + x2 = 5.678908346 Solving -7.5 + x2 = 5.678908346 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + x2 = 5.678908346 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + x2 = 5.678908346 + 7.5 x2 = 5.678908346 + 7.5 Combine like terms: 5.678908346 + 7.5 = 13.178908346 x2 = 13.178908346 Simplifying x2 = 13.178908346 Take the square root of each side: x = {-3.630276621, 3.630276621}

Subproblem 2

x2 + -7.5 = -5.678908346 Simplifying x2 + -7.5 = -5.678908346 Reorder the terms: -7.5 + x2 = -5.678908346 Solving -7.5 + x2 = -5.678908346 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + x2 = -5.678908346 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + x2 = -5.678908346 + 7.5 x2 = -5.678908346 + 7.5 Combine like terms: -5.678908346 + 7.5 = 1.821091654 x2 = 1.821091654 Simplifying x2 = 1.821091654 Take the square root of each side: x = {-1.34947829, 1.34947829}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-3.630276621, 3.630276621, -1.34947829, 1.34947829}

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