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Simplifying p2 + 6p = -1 Reorder the terms: 6p + p2 = -1 Solving 6p + p2 = -1 Solving for variable 'p'. Reorder the terms: 1 + 6p + p2 = -1 + 1 Combine like terms: -1 + 1 = 0 1 + 6p + p2 = 0 Begin completing the square. Move the constant term to the right: Add '-1' to each side of the equation. 1 + 6p + -1 + p2 = 0 + -1 Reorder the terms: 1 + -1 + 6p + p2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + 6p + p2 = 0 + -1 6p + p2 = 0 + -1 Combine like terms: 0 + -1 = -1 6p + p2 = -1 The p term is 6p. Take half its coefficient (3). Square it (9) and add it to both sides. Add '9' to each side of the equation. 6p + 9 + p2 = -1 + 9 Reorder the terms: 9 + 6p + p2 = -1 + 9 Combine like terms: -1 + 9 = 8 9 + 6p + p2 = 8 Factor a perfect square on the left side: (p + 3)(p + 3) = 8 Calculate the square root of the right side: 2.828427125 Break this problem into two subproblems by setting (p + 3) equal to 2.828427125 and -2.828427125.Subproblem 1
p + 3 = 2.828427125 Simplifying p + 3 = 2.828427125 Reorder the terms: 3 + p = 2.828427125 Solving 3 + p = 2.828427125 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + p = 2.828427125 + -3 Combine like terms: 3 + -3 = 0 0 + p = 2.828427125 + -3 p = 2.828427125 + -3 Combine like terms: 2.828427125 + -3 = -0.171572875 p = -0.171572875 Simplifying p = -0.171572875Subproblem 2
p + 3 = -2.828427125 Simplifying p + 3 = -2.828427125 Reorder the terms: 3 + p = -2.828427125 Solving 3 + p = -2.828427125 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + p = -2.828427125 + -3 Combine like terms: 3 + -3 = 0 0 + p = -2.828427125 + -3 p = -2.828427125 + -3 Combine like terms: -2.828427125 + -3 = -5.828427125 p = -5.828427125 Simplifying p = -5.828427125Solution
The solution to the problem is based on the solutions from the subproblems. p = {-0.171572875, -5.828427125}
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