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Simplifying 0 = 2p2 + -4p + -135 Reorder the terms: 0 = -135 + -4p + 2p2 Solving 0 = -135 + -4p + 2p2 Solving for variable 'p'. Combine like terms: 0 + 135 = 135 135 + 4p + -2p2 = -135 + -4p + 2p2 + 135 + 4p + -2p2 Reorder the terms: 135 + 4p + -2p2 = -135 + 135 + -4p + 4p + 2p2 + -2p2 Combine like terms: -135 + 135 = 0 135 + 4p + -2p2 = 0 + -4p + 4p + 2p2 + -2p2 135 + 4p + -2p2 = -4p + 4p + 2p2 + -2p2 Combine like terms: -4p + 4p = 0 135 + 4p + -2p2 = 0 + 2p2 + -2p2 135 + 4p + -2p2 = 2p2 + -2p2 Combine like terms: 2p2 + -2p2 = 0 135 + 4p + -2p2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -67.5 + -2p + p2 = 0 Move the constant term to the right: Add '67.5' to each side of the equation. -67.5 + -2p + 67.5 + p2 = 0 + 67.5 Reorder the terms: -67.5 + 67.5 + -2p + p2 = 0 + 67.5 Combine like terms: -67.5 + 67.5 = 0.0 0.0 + -2p + p2 = 0 + 67.5 -2p + p2 = 0 + 67.5 Combine like terms: 0 + 67.5 = 67.5 -2p + p2 = 67.5 The p term is -2p. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2p + 1 + p2 = 67.5 + 1 Reorder the terms: 1 + -2p + p2 = 67.5 + 1 Combine like terms: 67.5 + 1 = 68.5 1 + -2p + p2 = 68.5 Factor a perfect square on the left side: (p + -1)(p + -1) = 68.5 Calculate the square root of the right side: 8.276472679 Break this problem into two subproblems by setting (p + -1) equal to 8.276472679 and -8.276472679.Subproblem 1
p + -1 = 8.276472679 Simplifying p + -1 = 8.276472679 Reorder the terms: -1 + p = 8.276472679 Solving -1 + p = 8.276472679 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + p = 8.276472679 + 1 Combine like terms: -1 + 1 = 0 0 + p = 8.276472679 + 1 p = 8.276472679 + 1 Combine like terms: 8.276472679 + 1 = 9.276472679 p = 9.276472679 Simplifying p = 9.276472679Subproblem 2
p + -1 = -8.276472679 Simplifying p + -1 = -8.276472679 Reorder the terms: -1 + p = -8.276472679 Solving -1 + p = -8.276472679 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + p = -8.276472679 + 1 Combine like terms: -1 + 1 = 0 0 + p = -8.276472679 + 1 p = -8.276472679 + 1 Combine like terms: -8.276472679 + 1 = -7.276472679 p = -7.276472679 Simplifying p = -7.276472679Solution
The solution to the problem is based on the solutions from the subproblems. p = {9.276472679, -7.276472679}
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