x^4+1=7x^2

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Solution for x^4+1=7x^2 equation:


Simplifying
x4 + 1 = 7x2

Reorder the terms:
1 + x4 = 7x2

Solving
1 + x4 = 7x2

Solving for variable 'x'.

Reorder the terms:
1 + -7x2 + x4 = 7x2 + -7x2

Combine like terms: 7x2 + -7x2 = 0
1 + -7x2 + x4 = 0

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -7x2 + -1 + x4 = 0 + -1

Reorder the terms:
1 + -1 + -7x2 + x4 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -7x2 + x4 = 0 + -1
-7x2 + x4 = 0 + -1

Combine like terms: 0 + -1 = -1
-7x2 + x4 = -1

The x term is -7x2.  Take half its coefficient (-3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
-7x2 + 12.25 + x4 = -1 + 12.25

Reorder the terms:
12.25 + -7x2 + x4 = -1 + 12.25

Combine like terms: -1 + 12.25 = 11.25
12.25 + -7x2 + x4 = 11.25

Factor a perfect square on the left side:
(x2 + -3.5)(x2 + -3.5) = 11.25

Calculate the square root of the right side: 3.354101966

Break this problem into two subproblems by setting 
(x2 + -3.5) equal to 3.354101966 and -3.354101966.

Subproblem 1

x2 + -3.5 = 3.354101966 Simplifying x2 + -3.5 = 3.354101966 Reorder the terms: -3.5 + x2 = 3.354101966 Solving -3.5 + x2 = 3.354101966 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + x2 = 3.354101966 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x2 = 3.354101966 + 3.5 x2 = 3.354101966 + 3.5 Combine like terms: 3.354101966 + 3.5 = 6.854101966 x2 = 6.854101966 Simplifying x2 = 6.854101966 Take the square root of each side: x = {-2.618033989, 2.618033989}

Subproblem 2

x2 + -3.5 = -3.354101966 Simplifying x2 + -3.5 = -3.354101966 Reorder the terms: -3.5 + x2 = -3.354101966 Solving -3.5 + x2 = -3.354101966 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + x2 = -3.354101966 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x2 = -3.354101966 + 3.5 x2 = -3.354101966 + 3.5 Combine like terms: -3.354101966 + 3.5 = 0.145898034 x2 = 0.145898034 Simplifying x2 = 0.145898034 Take the square root of each side: x = {-0.381966012, 0.381966012}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-2.618033989, 2.618033989, -0.381966012, 0.381966012}

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