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Simplifying p2 + -4p + 6p + -7 = 0 Reorder the terms: -7 + -4p + 6p + p2 = 0 Combine like terms: -4p + 6p = 2p -7 + 2p + p2 = 0 Solving -7 + 2p + p2 = 0 Solving for variable 'p'. 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 = 1.828427125 p = 1.828427125 Simplifying p = 1.828427125Subproblem 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 = -3.828427125 p = -3.828427125 Simplifying p = -3.828427125Solution
The solution to the problem is based on the solutions from the subproblems. p = {1.828427125, -3.828427125}
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