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Simplifying 14 = -1p(p + -8) Reorder the terms: 14 = -1p(-8 + p) 14 = (-8 * -1p + p * -1p) 14 = (8p + -1p2) Solving 14 = 8p + -1p2 Solving for variable 'p'. Reorder the terms: 14 + -8p + p2 = 8p + -8p + -1p2 + p2 Combine like terms: 8p + -8p = 0 14 + -8p + p2 = 0 + -1p2 + p2 14 + -8p + p2 = -1p2 + p2 Combine like terms: -1p2 + p2 = 0 14 + -8p + p2 = 0 Begin completing the square. Move the constant term to the right: Add '-14' to each side of the equation. 14 + -8p + -14 + p2 = 0 + -14 Reorder the terms: 14 + -14 + -8p + p2 = 0 + -14 Combine like terms: 14 + -14 = 0 0 + -8p + p2 = 0 + -14 -8p + p2 = 0 + -14 Combine like terms: 0 + -14 = -14 -8p + p2 = -14 The p term is -8p. Take half its coefficient (-4). Square it (16) and add it to both sides. Add '16' to each side of the equation. -8p + 16 + p2 = -14 + 16 Reorder the terms: 16 + -8p + p2 = -14 + 16 Combine like terms: -14 + 16 = 2 16 + -8p + p2 = 2 Factor a perfect square on the left side: (p + -4)(p + -4) = 2 Calculate the square root of the right side: 1.414213562 Break this problem into two subproblems by setting (p + -4) equal to 1.414213562 and -1.414213562.Subproblem 1
p + -4 = 1.414213562 Simplifying p + -4 = 1.414213562 Reorder the terms: -4 + p = 1.414213562 Solving -4 + p = 1.414213562 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + p = 1.414213562 + 4 Combine like terms: -4 + 4 = 0 0 + p = 1.414213562 + 4 p = 1.414213562 + 4 Combine like terms: 1.414213562 + 4 = 5.414213562 p = 5.414213562 Simplifying p = 5.414213562Subproblem 2
p + -4 = -1.414213562 Simplifying p + -4 = -1.414213562 Reorder the terms: -4 + p = -1.414213562 Solving -4 + p = -1.414213562 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + p = -1.414213562 + 4 Combine like terms: -4 + 4 = 0 0 + p = -1.414213562 + 4 p = -1.414213562 + 4 Combine like terms: -1.414213562 + 4 = 2.585786438 p = 2.585786438 Simplifying p = 2.585786438Solution
The solution to the problem is based on the solutions from the subproblems. p = {5.414213562, 2.585786438}
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