p^4-7p^2+1=0

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Solution for p^4-7p^2+1=0 equation:


Simplifying
p4 + -7p2 + 1 = 0

Reorder the terms:
1 + -7p2 + p4 = 0

Solving
1 + -7p2 + p4 = 0

Solving for variable 'p'.

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -7p2 + -1 + p4 = 0 + -1

Reorder the terms:
1 + -1 + -7p2 + p4 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -7p2 + p4 = 0 + -1
-7p2 + p4 = 0 + -1

Combine like terms: 0 + -1 = -1
-7p2 + p4 = -1

The p term is -7p2.  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.
-7p2 + 12.25 + p4 = -1 + 12.25

Reorder the terms:
12.25 + -7p2 + p4 = -1 + 12.25

Combine like terms: -1 + 12.25 = 11.25
12.25 + -7p2 + p4 = 11.25

Factor a perfect square on the left side:
(p2 + -3.5)(p2 + -3.5) = 11.25

Calculate the square root of the right side: 3.354101966

Break this problem into two subproblems by setting 
(p2 + -3.5) equal to 3.354101966 and -3.354101966.

Subproblem 1

p2 + -3.5 = 3.354101966 Simplifying p2 + -3.5 = 3.354101966 Reorder the terms: -3.5 + p2 = 3.354101966 Solving -3.5 + p2 = 3.354101966 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + p2 = 3.354101966 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + p2 = 3.354101966 + 3.5 p2 = 3.354101966 + 3.5 Combine like terms: 3.354101966 + 3.5 = 6.854101966 p2 = 6.854101966 Simplifying p2 = 6.854101966 Take the square root of each side: p = {-2.618033989, 2.618033989}

Subproblem 2

p2 + -3.5 = -3.354101966 Simplifying p2 + -3.5 = -3.354101966 Reorder the terms: -3.5 + p2 = -3.354101966 Solving -3.5 + p2 = -3.354101966 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + p2 = -3.354101966 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + p2 = -3.354101966 + 3.5 p2 = -3.354101966 + 3.5 Combine like terms: -3.354101966 + 3.5 = 0.145898034 p2 = 0.145898034 Simplifying p2 = 0.145898034 Take the square root of each side: p = {-0.381966012, 0.381966012}

Solution

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

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