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Simplifying 7 + 7p(p + -6) = -38 + 6p Reorder the terms: 7 + 7p(-6 + p) = -38 + 6p 7 + (-6 * 7p + p * 7p) = -38 + 6p 7 + (-42p + 7p2) = -38 + 6p Solving 7 + -42p + 7p2 = -38 + 6p Solving for variable 'p'. Reorder the terms: 7 + 38 + -42p + -6p + 7p2 = -38 + 6p + 38 + -6p Combine like terms: 7 + 38 = 45 45 + -42p + -6p + 7p2 = -38 + 6p + 38 + -6p Combine like terms: -42p + -6p = -48p 45 + -48p + 7p2 = -38 + 6p + 38 + -6p Reorder the terms: 45 + -48p + 7p2 = -38 + 38 + 6p + -6p Combine like terms: -38 + 38 = 0 45 + -48p + 7p2 = 0 + 6p + -6p 45 + -48p + 7p2 = 6p + -6p Combine like terms: 6p + -6p = 0 45 + -48p + 7p2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. 6.428571429 + -6.857142857p + p2 = 0 Move the constant term to the right: Add '-6.428571429' to each side of the equation. 6.428571429 + -6.857142857p + -6.428571429 + p2 = 0 + -6.428571429 Reorder the terms: 6.428571429 + -6.428571429 + -6.857142857p + p2 = 0 + -6.428571429 Combine like terms: 6.428571429 + -6.428571429 = 0.000000000 0.000000000 + -6.857142857p + p2 = 0 + -6.428571429 -6.857142857p + p2 = 0 + -6.428571429 Combine like terms: 0 + -6.428571429 = -6.428571429 -6.857142857p + p2 = -6.428571429 The p term is -6.857142857p. Take half its coefficient (-3.428571429). Square it (11.75510204) and add it to both sides. Add '11.75510204' to each side of the equation. -6.857142857p + 11.75510204 + p2 = -6.428571429 + 11.75510204 Reorder the terms: 11.75510204 + -6.857142857p + p2 = -6.428571429 + 11.75510204 Combine like terms: -6.428571429 + 11.75510204 = 5.326530611 11.75510204 + -6.857142857p + p2 = 5.326530611 Factor a perfect square on the left side: (p + -3.428571429)(p + -3.428571429) = 5.326530611 Calculate the square root of the right side: 2.307927774 Break this problem into two subproblems by setting (p + -3.428571429) equal to 2.307927774 and -2.307927774.Subproblem 1
p + -3.428571429 = 2.307927774 Simplifying p + -3.428571429 = 2.307927774 Reorder the terms: -3.428571429 + p = 2.307927774 Solving -3.428571429 + p = 2.307927774 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '3.428571429' to each side of the equation. -3.428571429 + 3.428571429 + p = 2.307927774 + 3.428571429 Combine like terms: -3.428571429 + 3.428571429 = 0.000000000 0.000000000 + p = 2.307927774 + 3.428571429 p = 2.307927774 + 3.428571429 Combine like terms: 2.307927774 + 3.428571429 = 5.736499203 p = 5.736499203 Simplifying p = 5.736499203Subproblem 2
p + -3.428571429 = -2.307927774 Simplifying p + -3.428571429 = -2.307927774 Reorder the terms: -3.428571429 + p = -2.307927774 Solving -3.428571429 + p = -2.307927774 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '3.428571429' to each side of the equation. -3.428571429 + 3.428571429 + p = -2.307927774 + 3.428571429 Combine like terms: -3.428571429 + 3.428571429 = 0.000000000 0.000000000 + p = -2.307927774 + 3.428571429 p = -2.307927774 + 3.428571429 Combine like terms: -2.307927774 + 3.428571429 = 1.120643655 p = 1.120643655 Simplifying p = 1.120643655Solution
The solution to the problem is based on the solutions from the subproblems. p = {5.736499203, 1.120643655}
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