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Simplifying 5p(p + 9) + -6(2 + 8p) = 0 Reorder the terms: 5p(9 + p) + -6(2 + 8p) = 0 (9 * 5p + p * 5p) + -6(2 + 8p) = 0 (45p + 5p2) + -6(2 + 8p) = 0 45p + 5p2 + (2 * -6 + 8p * -6) = 0 45p + 5p2 + (-12 + -48p) = 0 Reorder the terms: -12 + 45p + -48p + 5p2 = 0 Combine like terms: 45p + -48p = -3p -12 + -3p + 5p2 = 0 Solving -12 + -3p + 5p2 = 0 Solving for variable 'p'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -2.4 + -0.6p + p2 = 0 Move the constant term to the right: Add '2.4' to each side of the equation. -2.4 + -0.6p + 2.4 + p2 = 0 + 2.4 Reorder the terms: -2.4 + 2.4 + -0.6p + p2 = 0 + 2.4 Combine like terms: -2.4 + 2.4 = 0.0 0.0 + -0.6p + p2 = 0 + 2.4 -0.6p + p2 = 0 + 2.4 Combine like terms: 0 + 2.4 = 2.4 -0.6p + p2 = 2.4 The p term is -0.6p. Take half its coefficient (-0.3). Square it (0.09) and add it to both sides. Add '0.09' to each side of the equation. -0.6p + 0.09 + p2 = 2.4 + 0.09 Reorder the terms: 0.09 + -0.6p + p2 = 2.4 + 0.09 Combine like terms: 2.4 + 0.09 = 2.49 0.09 + -0.6p + p2 = 2.49 Factor a perfect square on the left side: (p + -0.3)(p + -0.3) = 2.49 Calculate the square root of the right side: 1.577973384 Break this problem into two subproblems by setting (p + -0.3) equal to 1.577973384 and -1.577973384.Subproblem 1
p + -0.3 = 1.577973384 Simplifying p + -0.3 = 1.577973384 Reorder the terms: -0.3 + p = 1.577973384 Solving -0.3 + p = 1.577973384 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.3' to each side of the equation. -0.3 + 0.3 + p = 1.577973384 + 0.3 Combine like terms: -0.3 + 0.3 = 0.0 0.0 + p = 1.577973384 + 0.3 p = 1.577973384 + 0.3 Combine like terms: 1.577973384 + 0.3 = 1.877973384 p = 1.877973384 Simplifying p = 1.877973384Subproblem 2
p + -0.3 = -1.577973384 Simplifying p + -0.3 = -1.577973384 Reorder the terms: -0.3 + p = -1.577973384 Solving -0.3 + p = -1.577973384 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.3' to each side of the equation. -0.3 + 0.3 + p = -1.577973384 + 0.3 Combine like terms: -0.3 + 0.3 = 0.0 0.0 + p = -1.577973384 + 0.3 p = -1.577973384 + 0.3 Combine like terms: -1.577973384 + 0.3 = -1.277973384 p = -1.277973384 Simplifying p = -1.277973384Solution
The solution to the problem is based on the solutions from the subproblems. p = {1.877973384, -1.277973384}
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