x^4-19x^2+7=0

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Solution for x^4-19x^2+7=0 equation:


Simplifying
x4 + -19x2 + 7 = 0

Reorder the terms:
7 + -19x2 + x4 = 0

Solving
7 + -19x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-7' to each side of the equation.
7 + -19x2 + -7 + x4 = 0 + -7

Reorder the terms:
7 + -7 + -19x2 + x4 = 0 + -7

Combine like terms: 7 + -7 = 0
0 + -19x2 + x4 = 0 + -7
-19x2 + x4 = 0 + -7

Combine like terms: 0 + -7 = -7
-19x2 + x4 = -7

The x term is -19x2.  Take half its coefficient (-9.5).
Square it (90.25) and add it to both sides.

Add '90.25' to each side of the equation.
-19x2 + 90.25 + x4 = -7 + 90.25

Reorder the terms:
90.25 + -19x2 + x4 = -7 + 90.25

Combine like terms: -7 + 90.25 = 83.25
90.25 + -19x2 + x4 = 83.25

Factor a perfect square on the left side:
(x2 + -9.5)(x2 + -9.5) = 83.25

Calculate the square root of the right side: 9.124143795

Break this problem into two subproblems by setting 
(x2 + -9.5) equal to 9.124143795 and -9.124143795.

Subproblem 1

x2 + -9.5 = 9.124143795 Simplifying x2 + -9.5 = 9.124143795 Reorder the terms: -9.5 + x2 = 9.124143795 Solving -9.5 + x2 = 9.124143795 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '9.5' to each side of the equation. -9.5 + 9.5 + x2 = 9.124143795 + 9.5 Combine like terms: -9.5 + 9.5 = 0.0 0.0 + x2 = 9.124143795 + 9.5 x2 = 9.124143795 + 9.5 Combine like terms: 9.124143795 + 9.5 = 18.624143795 x2 = 18.624143795 Simplifying x2 = 18.624143795 Take the square root of each side: x = {-4.315569927, 4.315569927}

Subproblem 2

x2 + -9.5 = -9.124143795 Simplifying x2 + -9.5 = -9.124143795 Reorder the terms: -9.5 + x2 = -9.124143795 Solving -9.5 + x2 = -9.124143795 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '9.5' to each side of the equation. -9.5 + 9.5 + x2 = -9.124143795 + 9.5 Combine like terms: -9.5 + 9.5 = 0.0 0.0 + x2 = -9.124143795 + 9.5 x2 = -9.124143795 + 9.5 Combine like terms: -9.124143795 + 9.5 = 0.375856205 x2 = 0.375856205 Simplifying x2 = 0.375856205 Take the square root of each side: x = {-0.613071126, 0.613071126}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-4.315569927, 4.315569927, -0.613071126, 0.613071126}

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