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Simplifying 3p + -5 = (p + -1)(p + -2) Reorder the terms: -5 + 3p = (p + -1)(p + -2) Reorder the terms: -5 + 3p = (-1 + p)(p + -2) Reorder the terms: -5 + 3p = (-1 + p)(-2 + p) Multiply (-1 + p) * (-2 + p) -5 + 3p = (-1(-2 + p) + p(-2 + p)) -5 + 3p = ((-2 * -1 + p * -1) + p(-2 + p)) -5 + 3p = ((2 + -1p) + p(-2 + p)) -5 + 3p = (2 + -1p + (-2 * p + p * p)) -5 + 3p = (2 + -1p + (-2p + p2)) Combine like terms: -1p + -2p = -3p -5 + 3p = (2 + -3p + p2) Solving -5 + 3p = 2 + -3p + p2 Solving for variable 'p'. Reorder the terms: -5 + -2 + 3p + 3p + -1p2 = 2 + -3p + p2 + -2 + 3p + -1p2 Combine like terms: -5 + -2 = -7 -7 + 3p + 3p + -1p2 = 2 + -3p + p2 + -2 + 3p + -1p2 Combine like terms: 3p + 3p = 6p -7 + 6p + -1p2 = 2 + -3p + p2 + -2 + 3p + -1p2 Reorder the terms: -7 + 6p + -1p2 = 2 + -2 + -3p + 3p + p2 + -1p2 Combine like terms: 2 + -2 = 0 -7 + 6p + -1p2 = 0 + -3p + 3p + p2 + -1p2 -7 + 6p + -1p2 = -3p + 3p + p2 + -1p2 Combine like terms: -3p + 3p = 0 -7 + 6p + -1p2 = 0 + p2 + -1p2 -7 + 6p + -1p2 = p2 + -1p2 Combine like terms: p2 + -1p2 = 0 -7 + 6p + -1p2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 7 + -6p + p2 = 0 Move the constant term to the right: Add '-7' to each side of the equation. 7 + -6p + -7 + p2 = 0 + -7 Reorder the terms: 7 + -7 + -6p + p2 = 0 + -7 Combine like terms: 7 + -7 = 0 0 + -6p + p2 = 0 + -7 -6p + p2 = 0 + -7 Combine like terms: 0 + -7 = -7 -6p + p2 = -7 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 = -7 + 9 Reorder the terms: 9 + -6p + p2 = -7 + 9 Combine like terms: -7 + 9 = 2 9 + -6p + p2 = 2 Factor a perfect square on the left side: (p + -3)(p + -3) = 2 Calculate the square root of the right side: 1.414213562 Break this problem into two subproblems by setting (p + -3) equal to 1.414213562 and -1.414213562.Subproblem 1
p + -3 = 1.414213562 Simplifying p + -3 = 1.414213562 Reorder the terms: -3 + p = 1.414213562 Solving -3 + p = 1.414213562 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 = 1.414213562 + 3 Combine like terms: -3 + 3 = 0 0 + p = 1.414213562 + 3 p = 1.414213562 + 3 Combine like terms: 1.414213562 + 3 = 4.414213562 p = 4.414213562 Simplifying p = 4.414213562Subproblem 2
p + -3 = -1.414213562 Simplifying p + -3 = -1.414213562 Reorder the terms: -3 + p = -1.414213562 Solving -3 + p = -1.414213562 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 = -1.414213562 + 3 Combine like terms: -3 + 3 = 0 0 + p = -1.414213562 + 3 p = -1.414213562 + 3 Combine like terms: -1.414213562 + 3 = 1.585786438 p = 1.585786438 Simplifying p = 1.585786438Solution
The solution to the problem is based on the solutions from the subproblems. p = {4.414213562, 1.585786438}
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