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Simplifying 3(2p + -1) + 10 = -1(3p + 4) * 7p Reorder the terms: 3(-1 + 2p) + 10 = -1(3p + 4) * 7p (-1 * 3 + 2p * 3) + 10 = -1(3p + 4) * 7p (-3 + 6p) + 10 = -1(3p + 4) * 7p Reorder the terms: -3 + 10 + 6p = -1(3p + 4) * 7p Combine like terms: -3 + 10 = 7 7 + 6p = -1(3p + 4) * 7p Reorder the terms: 7 + 6p = -1(4 + 3p) * 7p Reorder the terms for easier multiplication: 7 + 6p = -1 * 7p(4 + 3p) Multiply -1 * 7 7 + 6p = -7p(4 + 3p) 7 + 6p = (4 * -7p + 3p * -7p) 7 + 6p = (-28p + -21p2) Solving 7 + 6p = -28p + -21p2 Solving for variable 'p'. Combine like terms: 6p + 28p = 34p 7 + 34p + 21p2 = -28p + -21p2 + 28p + 21p2 Reorder the terms: 7 + 34p + 21p2 = -28p + 28p + -21p2 + 21p2 Combine like terms: -28p + 28p = 0 7 + 34p + 21p2 = 0 + -21p2 + 21p2 7 + 34p + 21p2 = -21p2 + 21p2 Combine like terms: -21p2 + 21p2 = 0 7 + 34p + 21p2 = 0 Begin completing the square. Divide all terms by 21 the coefficient of the squared term: Divide each side by '21'. 0.3333333333 + 1.619047619p + p2 = 0 Move the constant term to the right: Add '-0.3333333333' to each side of the equation. 0.3333333333 + 1.619047619p + -0.3333333333 + p2 = 0 + -0.3333333333 Reorder the terms: 0.3333333333 + -0.3333333333 + 1.619047619p + p2 = 0 + -0.3333333333 Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000 0.0000000000 + 1.619047619p + p2 = 0 + -0.3333333333 1.619047619p + p2 = 0 + -0.3333333333 Combine like terms: 0 + -0.3333333333 = -0.3333333333 1.619047619p + p2 = -0.3333333333 The p term is 1.619047619p. Take half its coefficient (0.8095238095). Square it (0.6553287981) and add it to both sides. Add '0.6553287981' to each side of the equation. 1.619047619p + 0.6553287981 + p2 = -0.3333333333 + 0.6553287981 Reorder the terms: 0.6553287981 + 1.619047619p + p2 = -0.3333333333 + 0.6553287981 Combine like terms: -0.3333333333 + 0.6553287981 = 0.3219954648 0.6553287981 + 1.619047619p + p2 = 0.3219954648 Factor a perfect square on the left side: (p + 0.8095238095)(p + 0.8095238095) = 0.3219954648 Calculate the square root of the right side: 0.567446442 Break this problem into two subproblems by setting (p + 0.8095238095) equal to 0.567446442 and -0.567446442.Subproblem 1
p + 0.8095238095 = 0.567446442 Simplifying p + 0.8095238095 = 0.567446442 Reorder the terms: 0.8095238095 + p = 0.567446442 Solving 0.8095238095 + p = 0.567446442 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.8095238095' to each side of the equation. 0.8095238095 + -0.8095238095 + p = 0.567446442 + -0.8095238095 Combine like terms: 0.8095238095 + -0.8095238095 = 0.0000000000 0.0000000000 + p = 0.567446442 + -0.8095238095 p = 0.567446442 + -0.8095238095 Combine like terms: 0.567446442 + -0.8095238095 = -0.2420773675 p = -0.2420773675 Simplifying p = -0.2420773675Subproblem 2
p + 0.8095238095 = -0.567446442 Simplifying p + 0.8095238095 = -0.567446442 Reorder the terms: 0.8095238095 + p = -0.567446442 Solving 0.8095238095 + p = -0.567446442 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.8095238095' to each side of the equation. 0.8095238095 + -0.8095238095 + p = -0.567446442 + -0.8095238095 Combine like terms: 0.8095238095 + -0.8095238095 = 0.0000000000 0.0000000000 + p = -0.567446442 + -0.8095238095 p = -0.567446442 + -0.8095238095 Combine like terms: -0.567446442 + -0.8095238095 = -1.3769702515 p = -1.3769702515 Simplifying p = -1.3769702515Solution
The solution to the problem is based on the solutions from the subproblems. p = {-0.2420773675, -1.3769702515}
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