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Simplifying (0.5p + -8)(0.25p + -1) = 0 Reorder the terms: (-8 + 0.5p)(0.25p + -1) = 0 Reorder the terms: (-8 + 0.5p)(-1 + 0.25p) = 0 Multiply (-8 + 0.5p) * (-1 + 0.25p) (-8(-1 + 0.25p) + 0.5p * (-1 + 0.25p)) = 0 ((-1 * -8 + 0.25p * -8) + 0.5p * (-1 + 0.25p)) = 0 ((8 + -2p) + 0.5p * (-1 + 0.25p)) = 0 (8 + -2p + (-1 * 0.5p + 0.25p * 0.5p)) = 0 (8 + -2p + (-0.5p + 0.125p2)) = 0 Combine like terms: -2p + -0.5p = -2.5p (8 + -2.5p + 0.125p2) = 0 Solving 8 + -2.5p + 0.125p2 = 0 Solving for variable 'p'. Begin completing the square. Divide all terms by 0.125 the coefficient of the squared term: Divide each side by '0.125'. 64 + -20p + p2 = 0 Move the constant term to the right: Add '-64' to each side of the equation. 64 + -20p + -64 + p2 = 0 + -64 Reorder the terms: 64 + -64 + -20p + p2 = 0 + -64 Combine like terms: 64 + -64 = 0 0 + -20p + p2 = 0 + -64 -20p + p2 = 0 + -64 Combine like terms: 0 + -64 = -64 -20p + p2 = -64 The p term is -20p. Take half its coefficient (-10). Square it (100) and add it to both sides. Add '100' to each side of the equation. -20p + 100 + p2 = -64 + 100 Reorder the terms: 100 + -20p + p2 = -64 + 100 Combine like terms: -64 + 100 = 36 100 + -20p + p2 = 36 Factor a perfect square on the left side: (p + -10)(p + -10) = 36 Calculate the square root of the right side: 6 Break this problem into two subproblems by setting (p + -10) equal to 6 and -6.Subproblem 1
p + -10 = 6 Simplifying p + -10 = 6 Reorder the terms: -10 + p = 6 Solving -10 + p = 6 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '10' to each side of the equation. -10 + 10 + p = 6 + 10 Combine like terms: -10 + 10 = 0 0 + p = 6 + 10 p = 6 + 10 Combine like terms: 6 + 10 = 16 p = 16 Simplifying p = 16Subproblem 2
p + -10 = -6 Simplifying p + -10 = -6 Reorder the terms: -10 + p = -6 Solving -10 + p = -6 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '10' to each side of the equation. -10 + 10 + p = -6 + 10 Combine like terms: -10 + 10 = 0 0 + p = -6 + 10 p = -6 + 10 Combine like terms: -6 + 10 = 4 p = 4 Simplifying p = 4Solution
The solution to the problem is based on the solutions from the subproblems. p = {16, 4}
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