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Simplifying 7p(p + 1) = 9 + -1p Reorder the terms: 7p(1 + p) = 9 + -1p (1 * 7p + p * 7p) = 9 + -1p (7p + 7p2) = 9 + -1p Solving 7p + 7p2 = 9 + -1p Solving for variable 'p'. Reorder the terms: -9 + 7p + p + 7p2 = 9 + -1p + -9 + p Combine like terms: 7p + p = 8p -9 + 8p + 7p2 = 9 + -1p + -9 + p Reorder the terms: -9 + 8p + 7p2 = 9 + -9 + -1p + p Combine like terms: 9 + -9 = 0 -9 + 8p + 7p2 = 0 + -1p + p -9 + 8p + 7p2 = -1p + p Combine like terms: -1p + p = 0 -9 + 8p + 7p2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -1.285714286 + 1.142857143p + p2 = 0 Move the constant term to the right: Add '1.285714286' to each side of the equation. -1.285714286 + 1.142857143p + 1.285714286 + p2 = 0 + 1.285714286 Reorder the terms: -1.285714286 + 1.285714286 + 1.142857143p + p2 = 0 + 1.285714286 Combine like terms: -1.285714286 + 1.285714286 = 0.000000000 0.000000000 + 1.142857143p + p2 = 0 + 1.285714286 1.142857143p + p2 = 0 + 1.285714286 Combine like terms: 0 + 1.285714286 = 1.285714286 1.142857143p + p2 = 1.285714286 The p term is 1.142857143p. Take half its coefficient (0.5714285715). Square it (0.3265306123) and add it to both sides. Add '0.3265306123' to each side of the equation. 1.142857143p + 0.3265306123 + p2 = 1.285714286 + 0.3265306123 Reorder the terms: 0.3265306123 + 1.142857143p + p2 = 1.285714286 + 0.3265306123 Combine like terms: 1.285714286 + 0.3265306123 = 1.6122448983 0.3265306123 + 1.142857143p + p2 = 1.6122448983 Factor a perfect square on the left side: (p + 0.5714285715)(p + 0.5714285715) = 1.6122448983 Calculate the square root of the right side: 1.26974206 Break this problem into two subproblems by setting (p + 0.5714285715) equal to 1.26974206 and -1.26974206.Subproblem 1
p + 0.5714285715 = 1.26974206 Simplifying p + 0.5714285715 = 1.26974206 Reorder the terms: 0.5714285715 + p = 1.26974206 Solving 0.5714285715 + p = 1.26974206 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.5714285715' to each side of the equation. 0.5714285715 + -0.5714285715 + p = 1.26974206 + -0.5714285715 Combine like terms: 0.5714285715 + -0.5714285715 = 0.0000000000 0.0000000000 + p = 1.26974206 + -0.5714285715 p = 1.26974206 + -0.5714285715 Combine like terms: 1.26974206 + -0.5714285715 = 0.6983134885 p = 0.6983134885 Simplifying p = 0.6983134885Subproblem 2
p + 0.5714285715 = -1.26974206 Simplifying p + 0.5714285715 = -1.26974206 Reorder the terms: 0.5714285715 + p = -1.26974206 Solving 0.5714285715 + p = -1.26974206 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.5714285715' to each side of the equation. 0.5714285715 + -0.5714285715 + p = -1.26974206 + -0.5714285715 Combine like terms: 0.5714285715 + -0.5714285715 = 0.0000000000 0.0000000000 + p = -1.26974206 + -0.5714285715 p = -1.26974206 + -0.5714285715 Combine like terms: -1.26974206 + -0.5714285715 = -1.8411706315 p = -1.8411706315 Simplifying p = -1.8411706315Solution
The solution to the problem is based on the solutions from the subproblems. p = {0.6983134885, -1.8411706315}
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