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Simplifying 13p2 + 81p + -196 = 0 Reorder the terms: -196 + 81p + 13p2 = 0 Solving -196 + 81p + 13p2 = 0 Solving for variable 'p'. Begin completing the square. Divide all terms by 13 the coefficient of the squared term: Divide each side by '13'. -15.07692308 + 6.230769231p + p2 = 0 Move the constant term to the right: Add '15.07692308' to each side of the equation. -15.07692308 + 6.230769231p + 15.07692308 + p2 = 0 + 15.07692308 Reorder the terms: -15.07692308 + 15.07692308 + 6.230769231p + p2 = 0 + 15.07692308 Combine like terms: -15.07692308 + 15.07692308 = 0.00000000 0.00000000 + 6.230769231p + p2 = 0 + 15.07692308 6.230769231p + p2 = 0 + 15.07692308 Combine like terms: 0 + 15.07692308 = 15.07692308 6.230769231p + p2 = 15.07692308 The p term is 6.230769231p. Take half its coefficient (3.115384616). Square it (9.705621306) and add it to both sides. Add '9.705621306' to each side of the equation. 6.230769231p + 9.705621306 + p2 = 15.07692308 + 9.705621306 Reorder the terms: 9.705621306 + 6.230769231p + p2 = 15.07692308 + 9.705621306 Combine like terms: 15.07692308 + 9.705621306 = 24.782544386 9.705621306 + 6.230769231p + p2 = 24.782544386 Factor a perfect square on the left side: (p + 3.115384616)(p + 3.115384616) = 24.782544386 Calculate the square root of the right side: 4.978206945 Break this problem into two subproblems by setting (p + 3.115384616) equal to 4.978206945 and -4.978206945.Subproblem 1
p + 3.115384616 = 4.978206945 Simplifying p + 3.115384616 = 4.978206945 Reorder the terms: 3.115384616 + p = 4.978206945 Solving 3.115384616 + p = 4.978206945 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-3.115384616' to each side of the equation. 3.115384616 + -3.115384616 + p = 4.978206945 + -3.115384616 Combine like terms: 3.115384616 + -3.115384616 = 0.000000000 0.000000000 + p = 4.978206945 + -3.115384616 p = 4.978206945 + -3.115384616 Combine like terms: 4.978206945 + -3.115384616 = 1.862822329 p = 1.862822329 Simplifying p = 1.862822329Subproblem 2
p + 3.115384616 = -4.978206945 Simplifying p + 3.115384616 = -4.978206945 Reorder the terms: 3.115384616 + p = -4.978206945 Solving 3.115384616 + p = -4.978206945 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-3.115384616' to each side of the equation. 3.115384616 + -3.115384616 + p = -4.978206945 + -3.115384616 Combine like terms: 3.115384616 + -3.115384616 = 0.000000000 0.000000000 + p = -4.978206945 + -3.115384616 p = -4.978206945 + -3.115384616 Combine like terms: -4.978206945 + -3.115384616 = -8.093591561 p = -8.093591561 Simplifying p = -8.093591561Solution
The solution to the problem is based on the solutions from the subproblems. p = {1.862822329, -8.093591561}
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