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Simplifying 8p2 + 3 = 16p Reorder the terms: 3 + 8p2 = 16p Solving 3 + 8p2 = 16p Solving for variable 'p'. Reorder the terms: 3 + -16p + 8p2 = 16p + -16p Combine like terms: 16p + -16p = 0 3 + -16p + 8p2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 0.375 + -2p + p2 = 0 Move the constant term to the right: Add '-0.375' to each side of the equation. 0.375 + -2p + -0.375 + p2 = 0 + -0.375 Reorder the terms: 0.375 + -0.375 + -2p + p2 = 0 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + -2p + p2 = 0 + -0.375 -2p + p2 = 0 + -0.375 Combine like terms: 0 + -0.375 = -0.375 -2p + p2 = -0.375 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 = -0.375 + 1 Reorder the terms: 1 + -2p + p2 = -0.375 + 1 Combine like terms: -0.375 + 1 = 0.625 1 + -2p + p2 = 0.625 Factor a perfect square on the left side: (p + -1)(p + -1) = 0.625 Calculate the square root of the right side: 0.790569415 Break this problem into two subproblems by setting (p + -1) equal to 0.790569415 and -0.790569415.Subproblem 1
p + -1 = 0.790569415 Simplifying p + -1 = 0.790569415 Reorder the terms: -1 + p = 0.790569415 Solving -1 + p = 0.790569415 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 = 0.790569415 + 1 Combine like terms: -1 + 1 = 0 0 + p = 0.790569415 + 1 p = 0.790569415 + 1 Combine like terms: 0.790569415 + 1 = 1.790569415 p = 1.790569415 Simplifying p = 1.790569415Subproblem 2
p + -1 = -0.790569415 Simplifying p + -1 = -0.790569415 Reorder the terms: -1 + p = -0.790569415 Solving -1 + p = -0.790569415 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 = -0.790569415 + 1 Combine like terms: -1 + 1 = 0 0 + p = -0.790569415 + 1 p = -0.790569415 + 1 Combine like terms: -0.790569415 + 1 = 0.209430585 p = 0.209430585 Simplifying p = 0.209430585Solution
The solution to the problem is based on the solutions from the subproblems. p = {1.790569415, 0.209430585}
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