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Simplifying p(7p + -3) + 6(4p + -5) = 0 Reorder the terms: p(-3 + 7p) + 6(4p + -5) = 0 (-3 * p + 7p * p) + 6(4p + -5) = 0 (-3p + 7p2) + 6(4p + -5) = 0 Reorder the terms: -3p + 7p2 + 6(-5 + 4p) = 0 -3p + 7p2 + (-5 * 6 + 4p * 6) = 0 -3p + 7p2 + (-30 + 24p) = 0 Reorder the terms: -30 + -3p + 24p + 7p2 = 0 Combine like terms: -3p + 24p = 21p -30 + 21p + 7p2 = 0 Solving -30 + 21p + 7p2 = 0 Solving for variable 'p'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -4.285714286 + 3p + p2 = 0 Move the constant term to the right: Add '4.285714286' to each side of the equation. -4.285714286 + 3p + 4.285714286 + p2 = 0 + 4.285714286 Reorder the terms: -4.285714286 + 4.285714286 + 3p + p2 = 0 + 4.285714286 Combine like terms: -4.285714286 + 4.285714286 = 0.000000000 0.000000000 + 3p + p2 = 0 + 4.285714286 3p + p2 = 0 + 4.285714286 Combine like terms: 0 + 4.285714286 = 4.285714286 3p + p2 = 4.285714286 The p term is 3p. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3p + 2.25 + p2 = 4.285714286 + 2.25 Reorder the terms: 2.25 + 3p + p2 = 4.285714286 + 2.25 Combine like terms: 4.285714286 + 2.25 = 6.535714286 2.25 + 3p + p2 = 6.535714286 Factor a perfect square on the left side: (p + 1.5)(p + 1.5) = 6.535714286 Calculate the square root of the right side: 2.55650431 Break this problem into two subproblems by setting (p + 1.5) equal to 2.55650431 and -2.55650431.Subproblem 1
p + 1.5 = 2.55650431 Simplifying p + 1.5 = 2.55650431 Reorder the terms: 1.5 + p = 2.55650431 Solving 1.5 + p = 2.55650431 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + p = 2.55650431 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + p = 2.55650431 + -1.5 p = 2.55650431 + -1.5 Combine like terms: 2.55650431 + -1.5 = 1.05650431 p = 1.05650431 Simplifying p = 1.05650431Subproblem 2
p + 1.5 = -2.55650431 Simplifying p + 1.5 = -2.55650431 Reorder the terms: 1.5 + p = -2.55650431 Solving 1.5 + p = -2.55650431 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + p = -2.55650431 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + p = -2.55650431 + -1.5 p = -2.55650431 + -1.5 Combine like terms: -2.55650431 + -1.5 = -4.05650431 p = -4.05650431 Simplifying p = -4.05650431Solution
The solution to the problem is based on the solutions from the subproblems. p = {1.05650431, -4.05650431}
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