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Simplifying 7p2 + -12p + -24 = 0 Reorder the terms: -24 + -12p + 7p2 = 0 Solving -24 + -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'. -3.428571429 + -1.714285714p + p2 = 0 Move the constant term to the right: Add '3.428571429' to each side of the equation. -3.428571429 + -1.714285714p + 3.428571429 + p2 = 0 + 3.428571429 Reorder the terms: -3.428571429 + 3.428571429 + -1.714285714p + p2 = 0 + 3.428571429 Combine like terms: -3.428571429 + 3.428571429 = 0.000000000 0.000000000 + -1.714285714p + p2 = 0 + 3.428571429 -1.714285714p + p2 = 0 + 3.428571429 Combine like terms: 0 + 3.428571429 = 3.428571429 -1.714285714p + p2 = 3.428571429 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 = 3.428571429 + 0.7346938773 Reorder the terms: 0.7346938773 + -1.714285714p + p2 = 3.428571429 + 0.7346938773 Combine like terms: 3.428571429 + 0.7346938773 = 4.1632653063 0.7346938773 + -1.714285714p + p2 = 4.1632653063 Factor a perfect square on the left side: (p + -0.857142857)(p + -0.857142857) = 4.1632653063 Calculate the square root of the right side: 2.040408122 Break this problem into two subproblems by setting (p + -0.857142857) equal to 2.040408122 and -2.040408122.Subproblem 1
p + -0.857142857 = 2.040408122 Simplifying p + -0.857142857 = 2.040408122 Reorder the terms: -0.857142857 + p = 2.040408122 Solving -0.857142857 + p = 2.040408122 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 = 2.040408122 + 0.857142857 Combine like terms: -0.857142857 + 0.857142857 = 0.000000000 0.000000000 + p = 2.040408122 + 0.857142857 p = 2.040408122 + 0.857142857 Combine like terms: 2.040408122 + 0.857142857 = 2.897550979 p = 2.897550979 Simplifying p = 2.897550979Subproblem 2
p + -0.857142857 = -2.040408122 Simplifying p + -0.857142857 = -2.040408122 Reorder the terms: -0.857142857 + p = -2.040408122 Solving -0.857142857 + p = -2.040408122 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 = -2.040408122 + 0.857142857 Combine like terms: -0.857142857 + 0.857142857 = 0.000000000 0.000000000 + p = -2.040408122 + 0.857142857 p = -2.040408122 + 0.857142857 Combine like terms: -2.040408122 + 0.857142857 = -1.183265265 p = -1.183265265 Simplifying p = -1.183265265Solution
The solution to the problem is based on the solutions from the subproblems. p = {2.897550979, -1.183265265}
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