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Simplifying 8100 + -4p2 = p + -9 Reorder the terms: 8100 + -4p2 = -9 + p Solving 8100 + -4p2 = -9 + p Solving for variable 'p'. Reorder the terms: 8100 + 9 + -1p + -4p2 = -9 + p + 9 + -1p Combine like terms: 8100 + 9 = 8109 8109 + -1p + -4p2 = -9 + p + 9 + -1p Reorder the terms: 8109 + -1p + -4p2 = -9 + 9 + p + -1p Combine like terms: -9 + 9 = 0 8109 + -1p + -4p2 = 0 + p + -1p 8109 + -1p + -4p2 = p + -1p Combine like terms: p + -1p = 0 8109 + -1p + -4p2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -2027.25 + 0.25p + p2 = 0 Move the constant term to the right: Add '2027.25' to each side of the equation. -2027.25 + 0.25p + 2027.25 + p2 = 0 + 2027.25 Reorder the terms: -2027.25 + 2027.25 + 0.25p + p2 = 0 + 2027.25 Combine like terms: -2027.25 + 2027.25 = 0.00 0.00 + 0.25p + p2 = 0 + 2027.25 0.25p + p2 = 0 + 2027.25 Combine like terms: 0 + 2027.25 = 2027.25 0.25p + p2 = 2027.25 The p term is 0.25p. Take half its coefficient (0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. 0.25p + 0.015625 + p2 = 2027.25 + 0.015625 Reorder the terms: 0.015625 + 0.25p + p2 = 2027.25 + 0.015625 Combine like terms: 2027.25 + 0.015625 = 2027.265625 0.015625 + 0.25p + p2 = 2027.265625 Factor a perfect square on the left side: (p + 0.125)(p + 0.125) = 2027.265625 Calculate the square root of the right side: 45.025166574 Break this problem into two subproblems by setting (p + 0.125) equal to 45.025166574 and -45.025166574.Subproblem 1
p + 0.125 = 45.025166574 Simplifying p + 0.125 = 45.025166574 Reorder the terms: 0.125 + p = 45.025166574 Solving 0.125 + p = 45.025166574 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + p = 45.025166574 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + p = 45.025166574 + -0.125 p = 45.025166574 + -0.125 Combine like terms: 45.025166574 + -0.125 = 44.900166574 p = 44.900166574 Simplifying p = 44.900166574Subproblem 2
p + 0.125 = -45.025166574 Simplifying p + 0.125 = -45.025166574 Reorder the terms: 0.125 + p = -45.025166574 Solving 0.125 + p = -45.025166574 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + p = -45.025166574 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + p = -45.025166574 + -0.125 p = -45.025166574 + -0.125 Combine like terms: -45.025166574 + -0.125 = -45.150166574 p = -45.150166574 Simplifying p = -45.150166574Solution
The solution to the problem is based on the solutions from the subproblems. p = {44.900166574, -45.150166574}
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