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Simplifying 3 + p + 2(p + -5) * p = 5 Reorder the terms: 3 + p + 2(-5 + p) * p = 5 Reorder the terms for easier multiplication: 3 + p + 2p(-5 + p) = 5 3 + p + (-5 * 2p + p * 2p) = 5 3 + p + (-10p + 2p2) = 5 Combine like terms: p + -10p = -9p 3 + -9p + 2p2 = 5 Solving 3 + -9p + 2p2 = 5 Solving for variable 'p'. Reorder the terms: 3 + -5 + -9p + 2p2 = 5 + -5 Combine like terms: 3 + -5 = -2 -2 + -9p + 2p2 = 5 + -5 Combine like terms: 5 + -5 = 0 -2 + -9p + 2p2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -1 + -4.5p + p2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -4.5p + 1 + p2 = 0 + 1 Reorder the terms: -1 + 1 + -4.5p + p2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -4.5p + p2 = 0 + 1 -4.5p + p2 = 0 + 1 Combine like terms: 0 + 1 = 1 -4.5p + p2 = 1 The p term is -4.5p. Take half its coefficient (-2.25). Square it (5.0625) and add it to both sides. Add '5.0625' to each side of the equation. -4.5p + 5.0625 + p2 = 1 + 5.0625 Reorder the terms: 5.0625 + -4.5p + p2 = 1 + 5.0625 Combine like terms: 1 + 5.0625 = 6.0625 5.0625 + -4.5p + p2 = 6.0625 Factor a perfect square on the left side: (p + -2.25)(p + -2.25) = 6.0625 Calculate the square root of the right side: 2.46221445 Break this problem into two subproblems by setting (p + -2.25) equal to 2.46221445 and -2.46221445.Subproblem 1
p + -2.25 = 2.46221445 Simplifying p + -2.25 = 2.46221445 Reorder the terms: -2.25 + p = 2.46221445 Solving -2.25 + p = 2.46221445 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + p = 2.46221445 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + p = 2.46221445 + 2.25 p = 2.46221445 + 2.25 Combine like terms: 2.46221445 + 2.25 = 4.71221445 p = 4.71221445 Simplifying p = 4.71221445Subproblem 2
p + -2.25 = -2.46221445 Simplifying p + -2.25 = -2.46221445 Reorder the terms: -2.25 + p = -2.46221445 Solving -2.25 + p = -2.46221445 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + p = -2.46221445 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + p = -2.46221445 + 2.25 p = -2.46221445 + 2.25 Combine like terms: -2.46221445 + 2.25 = -0.21221445 p = -0.21221445 Simplifying p = -0.21221445Solution
The solution to the problem is based on the solutions from the subproblems. p = {4.71221445, -0.21221445}
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