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Simplifying -0.18p + -0.05(3 + -5p) = 0.04p(p + -4) + -0.26 -0.18p + (3 * -0.05 + -5p * -0.05) = 0.04p(p + -4) + -0.26 -0.18p + (-0.15 + 0.25p) = 0.04p(p + -4) + -0.26 Reorder the terms: -0.15 + -0.18p + 0.25p = 0.04p(p + -4) + -0.26 Combine like terms: -0.18p + 0.25p = 0.07p -0.15 + 0.07p = 0.04p(p + -4) + -0.26 Reorder the terms: -0.15 + 0.07p = 0.04p(-4 + p) + -0.26 -0.15 + 0.07p = (-4 * 0.04p + p * 0.04p) + -0.26 -0.15 + 0.07p = (-0.16p + 0.04p2) + -0.26 Reorder the terms: -0.15 + 0.07p = -0.26 + -0.16p + 0.04p2 Solving -0.15 + 0.07p = -0.26 + -0.16p + 0.04p2 Solving for variable 'p'. Reorder the terms: -0.15 + 0.26 + 0.07p + 0.16p + -0.04p2 = -0.26 + -0.16p + 0.04p2 + 0.26 + 0.16p + -0.04p2 Combine like terms: -0.15 + 0.26 = 0.11 0.11 + 0.07p + 0.16p + -0.04p2 = -0.26 + -0.16p + 0.04p2 + 0.26 + 0.16p + -0.04p2 Combine like terms: 0.07p + 0.16p = 0.23p 0.11 + 0.23p + -0.04p2 = -0.26 + -0.16p + 0.04p2 + 0.26 + 0.16p + -0.04p2 Reorder the terms: 0.11 + 0.23p + -0.04p2 = -0.26 + 0.26 + -0.16p + 0.16p + 0.04p2 + -0.04p2 Combine like terms: -0.26 + 0.26 = 0.00 0.11 + 0.23p + -0.04p2 = 0.00 + -0.16p + 0.16p + 0.04p2 + -0.04p2 0.11 + 0.23p + -0.04p2 = -0.16p + 0.16p + 0.04p2 + -0.04p2 Combine like terms: -0.16p + 0.16p = 0.00 0.11 + 0.23p + -0.04p2 = 0.00 + 0.04p2 + -0.04p2 0.11 + 0.23p + -0.04p2 = 0.04p2 + -0.04p2 Combine like terms: 0.04p2 + -0.04p2 = 0.00 0.11 + 0.23p + -0.04p2 = 0.00 Begin completing the square. Divide all terms by -0.04 the coefficient of the squared term: Divide each side by '-0.04'. -2.75 + -5.75p + p2 = 0 Move the constant term to the right: Add '2.75' to each side of the equation. -2.75 + -5.75p + 2.75 + p2 = 0 + 2.75 Reorder the terms: -2.75 + 2.75 + -5.75p + p2 = 0 + 2.75 Combine like terms: -2.75 + 2.75 = 0.00 0.00 + -5.75p + p2 = 0 + 2.75 -5.75p + p2 = 0 + 2.75 Combine like terms: 0 + 2.75 = 2.75 -5.75p + p2 = 2.75 The p term is -5.75p. Take half its coefficient (-2.875). Square it (8.265625) and add it to both sides. Add '8.265625' to each side of the equation. -5.75p + 8.265625 + p2 = 2.75 + 8.265625 Reorder the terms: 8.265625 + -5.75p + p2 = 2.75 + 8.265625 Combine like terms: 2.75 + 8.265625 = 11.015625 8.265625 + -5.75p + p2 = 11.015625 Factor a perfect square on the left side: (p + -2.875)(p + -2.875) = 11.015625 Calculate the square root of the right side: 3.318979512 Break this problem into two subproblems by setting (p + -2.875) equal to 3.318979512 and -3.318979512.Subproblem 1
p + -2.875 = 3.318979512 Simplifying p + -2.875 = 3.318979512 Reorder the terms: -2.875 + p = 3.318979512 Solving -2.875 + p = 3.318979512 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '2.875' to each side of the equation. -2.875 + 2.875 + p = 3.318979512 + 2.875 Combine like terms: -2.875 + 2.875 = 0.000 0.000 + p = 3.318979512 + 2.875 p = 3.318979512 + 2.875 Combine like terms: 3.318979512 + 2.875 = 6.193979512 p = 6.193979512 Simplifying p = 6.193979512Subproblem 2
p + -2.875 = -3.318979512 Simplifying p + -2.875 = -3.318979512 Reorder the terms: -2.875 + p = -3.318979512 Solving -2.875 + p = -3.318979512 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '2.875' to each side of the equation. -2.875 + 2.875 + p = -3.318979512 + 2.875 Combine like terms: -2.875 + 2.875 = 0.000 0.000 + p = -3.318979512 + 2.875 p = -3.318979512 + 2.875 Combine like terms: -3.318979512 + 2.875 = -0.443979512 p = -0.443979512 Simplifying p = -0.443979512Solution
The solution to the problem is based on the solutions from the subproblems. p = {6.193979512, -0.443979512}
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