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Simplifying p4 + 1 = 7p2 Reorder the terms: 1 + p4 = 7p2 Solving 1 + p4 = 7p2 Solving for variable 'p'. Reorder the terms: 1 + -7p2 + p4 = 7p2 + -7p2 Combine like terms: 7p2 + -7p2 = 0 1 + -7p2 + p4 = 0 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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