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Simplifying (9p2 + -4p + -6) + -1(-3p2 + 8p + 5) = 0 Reorder the terms: (-6 + -4p + 9p2) + -1(-3p2 + 8p + 5) = 0 Remove parenthesis around (-6 + -4p + 9p2) -6 + -4p + 9p2 + -1(-3p2 + 8p + 5) = 0 Reorder the terms: -6 + -4p + 9p2 + -1(5 + 8p + -3p2) = 0 -6 + -4p + 9p2 + (5 * -1 + 8p * -1 + -3p2 * -1) = 0 -6 + -4p + 9p2 + (-5 + -8p + 3p2) = 0 Reorder the terms: -6 + -5 + -4p + -8p + 9p2 + 3p2 = 0 Combine like terms: -6 + -5 = -11 -11 + -4p + -8p + 9p2 + 3p2 = 0 Combine like terms: -4p + -8p = -12p -11 + -12p + 9p2 + 3p2 = 0 Combine like terms: 9p2 + 3p2 = 12p2 -11 + -12p + 12p2 = 0 Solving -11 + -12p + 12p2 = 0 Solving for variable 'p'. Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -0.9166666667 + -1p + p2 = 0 Move the constant term to the right: Add '0.9166666667' to each side of the equation. -0.9166666667 + -1p + 0.9166666667 + p2 = 0 + 0.9166666667 Reorder the terms: -0.9166666667 + 0.9166666667 + -1p + p2 = 0 + 0.9166666667 Combine like terms: -0.9166666667 + 0.9166666667 = 0.0000000000 0.0000000000 + -1p + p2 = 0 + 0.9166666667 -1p + p2 = 0 + 0.9166666667 Combine like terms: 0 + 0.9166666667 = 0.9166666667 -1p + p2 = 0.9166666667 The p term is -1p. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1p + 0.25 + p2 = 0.9166666667 + 0.25 Reorder the terms: 0.25 + -1p + p2 = 0.9166666667 + 0.25 Combine like terms: 0.9166666667 + 0.25 = 1.1666666667 0.25 + -1p + p2 = 1.1666666667 Factor a perfect square on the left side: (p + -0.5)(p + -0.5) = 1.1666666667 Calculate the square root of the right side: 1.08012345 Break this problem into two subproblems by setting (p + -0.5) equal to 1.08012345 and -1.08012345.Subproblem 1
p + -0.5 = 1.08012345 Simplifying p + -0.5 = 1.08012345 Reorder the terms: -0.5 + p = 1.08012345 Solving -0.5 + p = 1.08012345 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + p = 1.08012345 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + p = 1.08012345 + 0.5 p = 1.08012345 + 0.5 Combine like terms: 1.08012345 + 0.5 = 1.58012345 p = 1.58012345 Simplifying p = 1.58012345Subproblem 2
p + -0.5 = -1.08012345 Simplifying p + -0.5 = -1.08012345 Reorder the terms: -0.5 + p = -1.08012345 Solving -0.5 + p = -1.08012345 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + p = -1.08012345 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + p = -1.08012345 + 0.5 p = -1.08012345 + 0.5 Combine like terms: -1.08012345 + 0.5 = -0.58012345 p = -0.58012345 Simplifying p = -0.58012345Solution
The solution to the problem is based on the solutions from the subproblems. p = {1.58012345, -0.58012345}
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