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Simplifying 15p2 + 24p + 8 = 0 Reorder the terms: 8 + 24p + 15p2 = 0 Solving 8 + 24p + 15p2 = 0 Solving for variable 'p'. Begin completing the square. Divide all terms by 15 the coefficient of the squared term: Divide each side by '15'. 0.5333333333 + 1.6p + p2 = 0 Move the constant term to the right: Add '-0.5333333333' to each side of the equation. 0.5333333333 + 1.6p + -0.5333333333 + p2 = 0 + -0.5333333333 Reorder the terms: 0.5333333333 + -0.5333333333 + 1.6p + p2 = 0 + -0.5333333333 Combine like terms: 0.5333333333 + -0.5333333333 = 0.0000000000 0.0000000000 + 1.6p + p2 = 0 + -0.5333333333 1.6p + p2 = 0 + -0.5333333333 Combine like terms: 0 + -0.5333333333 = -0.5333333333 1.6p + p2 = -0.5333333333 The p term is 1.6p. Take half its coefficient (0.8). Square it (0.64) and add it to both sides. Add '0.64' to each side of the equation. 1.6p + 0.64 + p2 = -0.5333333333 + 0.64 Reorder the terms: 0.64 + 1.6p + p2 = -0.5333333333 + 0.64 Combine like terms: -0.5333333333 + 0.64 = 0.1066666667 0.64 + 1.6p + p2 = 0.1066666667 Factor a perfect square on the left side: (p + 0.8)(p + 0.8) = 0.1066666667 Calculate the square root of the right side: 0.326598632 Break this problem into two subproblems by setting (p + 0.8) equal to 0.326598632 and -0.326598632.Subproblem 1
p + 0.8 = 0.326598632 Simplifying p + 0.8 = 0.326598632 Reorder the terms: 0.8 + p = 0.326598632 Solving 0.8 + p = 0.326598632 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.8' to each side of the equation. 0.8 + -0.8 + p = 0.326598632 + -0.8 Combine like terms: 0.8 + -0.8 = 0.0 0.0 + p = 0.326598632 + -0.8 p = 0.326598632 + -0.8 Combine like terms: 0.326598632 + -0.8 = -0.473401368 p = -0.473401368 Simplifying p = -0.473401368Subproblem 2
p + 0.8 = -0.326598632 Simplifying p + 0.8 = -0.326598632 Reorder the terms: 0.8 + p = -0.326598632 Solving 0.8 + p = -0.326598632 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.8' to each side of the equation. 0.8 + -0.8 + p = -0.326598632 + -0.8 Combine like terms: 0.8 + -0.8 = 0.0 0.0 + p = -0.326598632 + -0.8 p = -0.326598632 + -0.8 Combine like terms: -0.326598632 + -0.8 = -1.126598632 p = -1.126598632 Simplifying p = -1.126598632Solution
The solution to the problem is based on the solutions from the subproblems. p = {-0.473401368, -1.126598632}
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