x^4-16x^2+65=2x^2-6

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Solution for x^4-16x^2+65=2x^2-6 equation:


Simplifying
x4 + -16x2 + 65 = 2x2 + -6

Reorder the terms:
65 + -16x2 + x4 = 2x2 + -6

Reorder the terms:
65 + -16x2 + x4 = -6 + 2x2

Solving
65 + -16x2 + x4 = -6 + 2x2

Solving for variable 'x'.

Reorder the terms:
65 + 6 + -16x2 + -2x2 + x4 = -6 + 2x2 + 6 + -2x2

Combine like terms: 65 + 6 = 71
71 + -16x2 + -2x2 + x4 = -6 + 2x2 + 6 + -2x2

Combine like terms: -16x2 + -2x2 = -18x2
71 + -18x2 + x4 = -6 + 2x2 + 6 + -2x2

Reorder the terms:
71 + -18x2 + x4 = -6 + 6 + 2x2 + -2x2

Combine like terms: -6 + 6 = 0
71 + -18x2 + x4 = 0 + 2x2 + -2x2
71 + -18x2 + x4 = 2x2 + -2x2

Combine like terms: 2x2 + -2x2 = 0
71 + -18x2 + x4 = 0

Begin completing the square.

Move the constant term to the right:

Add '-71' to each side of the equation.
71 + -18x2 + -71 + x4 = 0 + -71

Reorder the terms:
71 + -71 + -18x2 + x4 = 0 + -71

Combine like terms: 71 + -71 = 0
0 + -18x2 + x4 = 0 + -71
-18x2 + x4 = 0 + -71

Combine like terms: 0 + -71 = -71
-18x2 + x4 = -71

The x term is -18x2.  Take half its coefficient (-9).
Square it (81) and add it to both sides.

Add '81' to each side of the equation.
-18x2 + 81 + x4 = -71 + 81

Reorder the terms:
81 + -18x2 + x4 = -71 + 81

Combine like terms: -71 + 81 = 10
81 + -18x2 + x4 = 10

Factor a perfect square on the left side:
(x2 + -9)(x2 + -9) = 10

Calculate the square root of the right side: 3.16227766

Break this problem into two subproblems by setting 
(x2 + -9) equal to 3.16227766 and -3.16227766.

Subproblem 1

x2 + -9 = 3.16227766 Simplifying x2 + -9 = 3.16227766 Reorder the terms: -9 + x2 = 3.16227766 Solving -9 + x2 = 3.16227766 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + x2 = 3.16227766 + 9 Combine like terms: -9 + 9 = 0 0 + x2 = 3.16227766 + 9 x2 = 3.16227766 + 9 Combine like terms: 3.16227766 + 9 = 12.16227766 x2 = 12.16227766 Simplifying x2 = 12.16227766 Take the square root of each side: x = {-3.487445721, 3.487445721}

Subproblem 2

x2 + -9 = -3.16227766 Simplifying x2 + -9 = -3.16227766 Reorder the terms: -9 + x2 = -3.16227766 Solving -9 + x2 = -3.16227766 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + x2 = -3.16227766 + 9 Combine like terms: -9 + 9 = 0 0 + x2 = -3.16227766 + 9 x2 = -3.16227766 + 9 Combine like terms: -3.16227766 + 9 = 5.83772234 x2 = 5.83772234 Simplifying x2 = 5.83772234 Take the square root of each side: x = {-2.416137898, 2.416137898}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-3.487445721, 3.487445721, -2.416137898, 2.416137898}

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