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Simplifying p(5p + -3) + 2(4p + -6) = 0 Reorder the terms: p(-3 + 5p) + 2(4p + -6) = 0 (-3 * p + 5p * p) + 2(4p + -6) = 0 (-3p + 5p2) + 2(4p + -6) = 0 Reorder the terms: -3p + 5p2 + 2(-6 + 4p) = 0 -3p + 5p2 + (-6 * 2 + 4p * 2) = 0 -3p + 5p2 + (-12 + 8p) = 0 Reorder the terms: -12 + -3p + 8p + 5p2 = 0 Combine like terms: -3p + 8p = 5p -12 + 5p + 5p2 = 0 Solving -12 + 5p + 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 + p + p2 = 0 Move the constant term to the right: Add '2.4' to each side of the equation. -2.4 + p + 2.4 + p2 = 0 + 2.4 Reorder the terms: -2.4 + 2.4 + p + p2 = 0 + 2.4 Combine like terms: -2.4 + 2.4 = 0.0 0.0 + p + p2 = 0 + 2.4 p + p2 = 0 + 2.4 Combine like terms: 0 + 2.4 = 2.4 p + p2 = 2.4 The p term is p. 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. + 0.25 + p2 = 2.4 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + p2 = 2.4 + 0.25 Combine like terms: 2.4 + 0.25 = 2.65 1.25 + p2 = 2.65 Factor a perfect square on the left side: (p + 0.5)(p + 0.5) = 2.65 Calculate the square root of the right side: 1.62788206 Break this problem into two subproblems by setting (p + 0.5) equal to 1.62788206 and -1.62788206.Subproblem 1
p + 0.5 = 1.62788206 Simplifying p + 0.5 = 1.62788206 Reorder the terms: 0.5 + p = 1.62788206 Solving 0.5 + p = 1.62788206 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 = 1.62788206 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + p = 1.62788206 + -0.5 p = 1.62788206 + -0.5 Combine like terms: 1.62788206 + -0.5 = 1.12788206 p = 1.12788206 Simplifying p = 1.12788206Subproblem 2
p + 0.5 = -1.62788206 Simplifying p + 0.5 = -1.62788206 Reorder the terms: 0.5 + p = -1.62788206 Solving 0.5 + p = -1.62788206 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 = -1.62788206 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + p = -1.62788206 + -0.5 p = -1.62788206 + -0.5 Combine like terms: -1.62788206 + -0.5 = -2.12788206 p = -2.12788206 Simplifying p = -2.12788206Solution
The solution to the problem is based on the solutions from the subproblems. p = {1.12788206, -2.12788206}
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