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Simplifying 0.05p + 0.08p(2p) = 741 Remove parenthesis around (2p) 0.05p + 0.08p * 2p = 741 Reorder the terms for easier multiplication: 0.05p + 0.08 * 2p * p = 741 Multiply 0.08 * 2 0.05p + 0.16p * p = 741 Multiply p * p 0.05p + 0.16p2 = 741 Solving 0.05p + 0.16p2 = 741 Solving for variable 'p'. Reorder the terms: -741 + 0.05p + 0.16p2 = 741 + -741 Combine like terms: 741 + -741 = 0 -741 + 0.05p + 0.16p2 = 0 Begin completing the square. Divide all terms by 0.16 the coefficient of the squared term: Divide each side by '0.16'. -4631.25 + 0.3125p + p2 = 0 Move the constant term to the right: Add '4631.25' to each side of the equation. -4631.25 + 0.3125p + 4631.25 + p2 = 0 + 4631.25 Reorder the terms: -4631.25 + 4631.25 + 0.3125p + p2 = 0 + 4631.25 Combine like terms: -4631.25 + 4631.25 = 0.00 0.00 + 0.3125p + p2 = 0 + 4631.25 0.3125p + p2 = 0 + 4631.25 Combine like terms: 0 + 4631.25 = 4631.25 0.3125p + p2 = 4631.25 The p term is 0.3125p. Take half its coefficient (0.15625). Square it (0.0244140625) and add it to both sides. Add '0.0244140625' to each side of the equation. 0.3125p + 0.0244140625 + p2 = 4631.25 + 0.0244140625 Reorder the terms: 0.0244140625 + 0.3125p + p2 = 4631.25 + 0.0244140625 Combine like terms: 4631.25 + 0.0244140625 = 4631.2744140625 0.0244140625 + 0.3125p + p2 = 4631.2744140625 Factor a perfect square on the left side: (p + 0.15625)(p + 0.15625) = 4631.2744140625 Calculate the square root of the right side: 68.053467318 Break this problem into two subproblems by setting (p + 0.15625) equal to 68.053467318 and -68.053467318.Subproblem 1
p + 0.15625 = 68.053467318 Simplifying p + 0.15625 = 68.053467318 Reorder the terms: 0.15625 + p = 68.053467318 Solving 0.15625 + p = 68.053467318 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.15625' to each side of the equation. 0.15625 + -0.15625 + p = 68.053467318 + -0.15625 Combine like terms: 0.15625 + -0.15625 = 0.00000 0.00000 + p = 68.053467318 + -0.15625 p = 68.053467318 + -0.15625 Combine like terms: 68.053467318 + -0.15625 = 67.897217318 p = 67.897217318 Simplifying p = 67.897217318Subproblem 2
p + 0.15625 = -68.053467318 Simplifying p + 0.15625 = -68.053467318 Reorder the terms: 0.15625 + p = -68.053467318 Solving 0.15625 + p = -68.053467318 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.15625' to each side of the equation. 0.15625 + -0.15625 + p = -68.053467318 + -0.15625 Combine like terms: 0.15625 + -0.15625 = 0.00000 0.00000 + p = -68.053467318 + -0.15625 p = -68.053467318 + -0.15625 Combine like terms: -68.053467318 + -0.15625 = -68.209717318 p = -68.209717318 Simplifying p = -68.209717318Solution
The solution to the problem is based on the solutions from the subproblems. p = {67.897217318, -68.209717318}
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