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Simplifying 2p2 = 4p + 14 Reorder the terms: 2p2 = 14 + 4p Solving 2p2 = 14 + 4p Solving for variable 'p'. Reorder the terms: -14 + -4p + 2p2 = 14 + 4p + -14 + -4p Reorder the terms: -14 + -4p + 2p2 = 14 + -14 + 4p + -4p Combine like terms: 14 + -14 = 0 -14 + -4p + 2p2 = 0 + 4p + -4p -14 + -4p + 2p2 = 4p + -4p Combine like terms: 4p + -4p = 0 -14 + -4p + 2p2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-7 + -2p + p2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-7 + -2p + p2)' equal to zero and attempt to solve: Simplifying -7 + -2p + p2 = 0 Solving -7 + -2p + p2 = 0 Begin completing the square. Move the constant term to the right: Add '7' to each side of the equation. -7 + -2p + 7 + p2 = 0 + 7 Reorder the terms: -7 + 7 + -2p + p2 = 0 + 7 Combine like terms: -7 + 7 = 0 0 + -2p + p2 = 0 + 7 -2p + p2 = 0 + 7 Combine like terms: 0 + 7 = 7 -2p + p2 = 7 The p term is -2p. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2p + 1 + p2 = 7 + 1 Reorder the terms: 1 + -2p + p2 = 7 + 1 Combine like terms: 7 + 1 = 8 1 + -2p + p2 = 8 Factor a perfect square on the left side: (p + -1)(p + -1) = 8 Calculate the square root of the right side: 2.828427125 Break this problem into two subproblems by setting (p + -1) equal to 2.828427125 and -2.828427125.Subproblem 1
p + -1 = 2.828427125 Simplifying p + -1 = 2.828427125 Reorder the terms: -1 + p = 2.828427125 Solving -1 + p = 2.828427125 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + p = 2.828427125 + 1 Combine like terms: -1 + 1 = 0 0 + p = 2.828427125 + 1 p = 2.828427125 + 1 Combine like terms: 2.828427125 + 1 = 3.828427125 p = 3.828427125 Simplifying p = 3.828427125Subproblem 2
p + -1 = -2.828427125 Simplifying p + -1 = -2.828427125 Reorder the terms: -1 + p = -2.828427125 Solving -1 + p = -2.828427125 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + p = -2.828427125 + 1 Combine like terms: -1 + 1 = 0 0 + p = -2.828427125 + 1 p = -2.828427125 + 1 Combine like terms: -2.828427125 + 1 = -1.828427125 p = -1.828427125 Simplifying p = -1.828427125Solution
The solution to the problem is based on the solutions from the subproblems. p = {3.828427125, -1.828427125}Solution
p = {3.828427125, -1.828427125}
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