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Simplifying p(7p + -5) + 2(4p + -6) = 0 Reorder the terms: p(-5 + 7p) + 2(4p + -6) = 0 (-5 * p + 7p * p) + 2(4p + -6) = 0 (-5p + 7p2) + 2(4p + -6) = 0 Reorder the terms: -5p + 7p2 + 2(-6 + 4p) = 0 -5p + 7p2 + (-6 * 2 + 4p * 2) = 0 -5p + 7p2 + (-12 + 8p) = 0 Reorder the terms: -12 + -5p + 8p + 7p2 = 0 Combine like terms: -5p + 8p = 3p -12 + 3p + 7p2 = 0 Solving -12 + 3p + 7p2 = 0 Solving for variable 'p'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -1.714285714 + 0.4285714286p + p2 = 0 Move the constant term to the right: Add '1.714285714' to each side of the equation. -1.714285714 + 0.4285714286p + 1.714285714 + p2 = 0 + 1.714285714 Reorder the terms: -1.714285714 + 1.714285714 + 0.4285714286p + p2 = 0 + 1.714285714 Combine like terms: -1.714285714 + 1.714285714 = 0.000000000 0.000000000 + 0.4285714286p + p2 = 0 + 1.714285714 0.4285714286p + p2 = 0 + 1.714285714 Combine like terms: 0 + 1.714285714 = 1.714285714 0.4285714286p + p2 = 1.714285714 The p term is 0.4285714286p. Take half its coefficient (0.2142857143). Square it (0.04591836735) and add it to both sides. Add '0.04591836735' to each side of the equation. 0.4285714286p + 0.04591836735 + p2 = 1.714285714 + 0.04591836735 Reorder the terms: 0.04591836735 + 0.4285714286p + p2 = 1.714285714 + 0.04591836735 Combine like terms: 1.714285714 + 0.04591836735 = 1.76020408135 0.04591836735 + 0.4285714286p + p2 = 1.76020408135 Factor a perfect square on the left side: (p + 0.2142857143)(p + 0.2142857143) = 1.76020408135 Calculate the square root of the right side: 1.32672683 Break this problem into two subproblems by setting (p + 0.2142857143) equal to 1.32672683 and -1.32672683.Subproblem 1
p + 0.2142857143 = 1.32672683 Simplifying p + 0.2142857143 = 1.32672683 Reorder the terms: 0.2142857143 + p = 1.32672683 Solving 0.2142857143 + p = 1.32672683 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.2142857143' to each side of the equation. 0.2142857143 + -0.2142857143 + p = 1.32672683 + -0.2142857143 Combine like terms: 0.2142857143 + -0.2142857143 = 0.0000000000 0.0000000000 + p = 1.32672683 + -0.2142857143 p = 1.32672683 + -0.2142857143 Combine like terms: 1.32672683 + -0.2142857143 = 1.1124411157 p = 1.1124411157 Simplifying p = 1.1124411157Subproblem 2
p + 0.2142857143 = -1.32672683 Simplifying p + 0.2142857143 = -1.32672683 Reorder the terms: 0.2142857143 + p = -1.32672683 Solving 0.2142857143 + p = -1.32672683 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.2142857143' to each side of the equation. 0.2142857143 + -0.2142857143 + p = -1.32672683 + -0.2142857143 Combine like terms: 0.2142857143 + -0.2142857143 = 0.0000000000 0.0000000000 + p = -1.32672683 + -0.2142857143 p = -1.32672683 + -0.2142857143 Combine like terms: -1.32672683 + -0.2142857143 = -1.5410125443 p = -1.5410125443 Simplifying p = -1.5410125443Solution
The solution to the problem is based on the solutions from the subproblems. p = {1.1124411157, -1.5410125443}
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