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Simplifying 15p2 + -2p + -1 = 7p2 + 6 + 3p Reorder the terms: -1 + -2p + 15p2 = 7p2 + 6 + 3p Reorder the terms: -1 + -2p + 15p2 = 6 + 3p + 7p2 Solving -1 + -2p + 15p2 = 6 + 3p + 7p2 Solving for variable 'p'. Reorder the terms: -1 + -6 + -2p + -3p + 15p2 + -7p2 = 6 + 3p + 7p2 + -6 + -3p + -7p2 Combine like terms: -1 + -6 = -7 -7 + -2p + -3p + 15p2 + -7p2 = 6 + 3p + 7p2 + -6 + -3p + -7p2 Combine like terms: -2p + -3p = -5p -7 + -5p + 15p2 + -7p2 = 6 + 3p + 7p2 + -6 + -3p + -7p2 Combine like terms: 15p2 + -7p2 = 8p2 -7 + -5p + 8p2 = 6 + 3p + 7p2 + -6 + -3p + -7p2 Reorder the terms: -7 + -5p + 8p2 = 6 + -6 + 3p + -3p + 7p2 + -7p2 Combine like terms: 6 + -6 = 0 -7 + -5p + 8p2 = 0 + 3p + -3p + 7p2 + -7p2 -7 + -5p + 8p2 = 3p + -3p + 7p2 + -7p2 Combine like terms: 3p + -3p = 0 -7 + -5p + 8p2 = 0 + 7p2 + -7p2 -7 + -5p + 8p2 = 7p2 + -7p2 Combine like terms: 7p2 + -7p2 = 0 -7 + -5p + 8p2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.875 + -0.625p + p2 = 0 Move the constant term to the right: Add '0.875' to each side of the equation. -0.875 + -0.625p + 0.875 + p2 = 0 + 0.875 Reorder the terms: -0.875 + 0.875 + -0.625p + p2 = 0 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + -0.625p + p2 = 0 + 0.875 -0.625p + p2 = 0 + 0.875 Combine like terms: 0 + 0.875 = 0.875 -0.625p + p2 = 0.875 The p term is -0.625p. Take half its coefficient (-0.3125). Square it (0.09765625) and add it to both sides. Add '0.09765625' to each side of the equation. -0.625p + 0.09765625 + p2 = 0.875 + 0.09765625 Reorder the terms: 0.09765625 + -0.625p + p2 = 0.875 + 0.09765625 Combine like terms: 0.875 + 0.09765625 = 0.97265625 0.09765625 + -0.625p + p2 = 0.97265625 Factor a perfect square on the left side: (p + -0.3125)(p + -0.3125) = 0.97265625 Calculate the square root of the right side: 0.986233365 Break this problem into two subproblems by setting (p + -0.3125) equal to 0.986233365 and -0.986233365.Subproblem 1
p + -0.3125 = 0.986233365 Simplifying p + -0.3125 = 0.986233365 Reorder the terms: -0.3125 + p = 0.986233365 Solving -0.3125 + p = 0.986233365 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.3125' to each side of the equation. -0.3125 + 0.3125 + p = 0.986233365 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + p = 0.986233365 + 0.3125 p = 0.986233365 + 0.3125 Combine like terms: 0.986233365 + 0.3125 = 1.298733365 p = 1.298733365 Simplifying p = 1.298733365Subproblem 2
p + -0.3125 = -0.986233365 Simplifying p + -0.3125 = -0.986233365 Reorder the terms: -0.3125 + p = -0.986233365 Solving -0.3125 + p = -0.986233365 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.3125' to each side of the equation. -0.3125 + 0.3125 + p = -0.986233365 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + p = -0.986233365 + 0.3125 p = -0.986233365 + 0.3125 Combine like terms: -0.986233365 + 0.3125 = -0.673733365 p = -0.673733365 Simplifying p = -0.673733365Solution
The solution to the problem is based on the solutions from the subproblems. p = {1.298733365, -0.673733365}
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