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Simplifying p(7p + -6) + 6(3p + -5) = 0 Reorder the terms: p(-6 + 7p) + 6(3p + -5) = 0 (-6 * p + 7p * p) + 6(3p + -5) = 0 (-6p + 7p2) + 6(3p + -5) = 0 Reorder the terms: -6p + 7p2 + 6(-5 + 3p) = 0 -6p + 7p2 + (-5 * 6 + 3p * 6) = 0 -6p + 7p2 + (-30 + 18p) = 0 Reorder the terms: -30 + -6p + 18p + 7p2 = 0 Combine like terms: -6p + 18p = 12p -30 + 12p + 7p2 = 0 Solving -30 + 12p + 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 + 1.714285714p + p2 = 0 Move the constant term to the right: Add '4.285714286' to each side of the equation. -4.285714286 + 1.714285714p + 4.285714286 + p2 = 0 + 4.285714286 Reorder the terms: -4.285714286 + 4.285714286 + 1.714285714p + p2 = 0 + 4.285714286 Combine like terms: -4.285714286 + 4.285714286 = 0.000000000 0.000000000 + 1.714285714p + p2 = 0 + 4.285714286 1.714285714p + p2 = 0 + 4.285714286 Combine like terms: 0 + 4.285714286 = 4.285714286 1.714285714p + p2 = 4.285714286 The p term is 1.714285714p. Take half its coefficient (0.857142857). Square it (0.7346938773) and add it to both sides. Add '0.7346938773' to each side of the equation. 1.714285714p + 0.7346938773 + p2 = 4.285714286 + 0.7346938773 Reorder the terms: 0.7346938773 + 1.714285714p + p2 = 4.285714286 + 0.7346938773 Combine like terms: 4.285714286 + 0.7346938773 = 5.0204081633 0.7346938773 + 1.714285714p + p2 = 5.0204081633 Factor a perfect square on the left side: (p + 0.857142857)(p + 0.857142857) = 5.0204081633 Calculate the square root of the right side: 2.240626734 Break this problem into two subproblems by setting (p + 0.857142857) equal to 2.240626734 and -2.240626734.Subproblem 1
p + 0.857142857 = 2.240626734 Simplifying p + 0.857142857 = 2.240626734 Reorder the terms: 0.857142857 + p = 2.240626734 Solving 0.857142857 + p = 2.240626734 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.857142857' to each side of the equation. 0.857142857 + -0.857142857 + p = 2.240626734 + -0.857142857 Combine like terms: 0.857142857 + -0.857142857 = 0.000000000 0.000000000 + p = 2.240626734 + -0.857142857 p = 2.240626734 + -0.857142857 Combine like terms: 2.240626734 + -0.857142857 = 1.383483877 p = 1.383483877 Simplifying p = 1.383483877Subproblem 2
p + 0.857142857 = -2.240626734 Simplifying p + 0.857142857 = -2.240626734 Reorder the terms: 0.857142857 + p = -2.240626734 Solving 0.857142857 + p = -2.240626734 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.857142857' to each side of the equation. 0.857142857 + -0.857142857 + p = -2.240626734 + -0.857142857 Combine like terms: 0.857142857 + -0.857142857 = 0.000000000 0.000000000 + p = -2.240626734 + -0.857142857 p = -2.240626734 + -0.857142857 Combine like terms: -2.240626734 + -0.857142857 = -3.097769591 p = -3.097769591 Simplifying p = -3.097769591Solution
The solution to the problem is based on the solutions from the subproblems. p = {1.383483877, -3.097769591}
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