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Simplifying 6p2 + 27p + 10 = 0 Reorder the terms: 10 + 27p + 6p2 = 0 Solving 10 + 27p + 6p2 = 0 Solving for variable 'p'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. 1.666666667 + 4.5p + p2 = 0 Move the constant term to the right: Add '-1.666666667' to each side of the equation. 1.666666667 + 4.5p + -1.666666667 + p2 = 0 + -1.666666667 Reorder the terms: 1.666666667 + -1.666666667 + 4.5p + p2 = 0 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + 4.5p + p2 = 0 + -1.666666667 4.5p + p2 = 0 + -1.666666667 Combine like terms: 0 + -1.666666667 = -1.666666667 4.5p + p2 = -1.666666667 The p term is 4.5p. Take half its coefficient (2.25). Square it (5.0625) and add it to both sides. Add '5.0625' to each side of the equation. 4.5p + 5.0625 + p2 = -1.666666667 + 5.0625 Reorder the terms: 5.0625 + 4.5p + p2 = -1.666666667 + 5.0625 Combine like terms: -1.666666667 + 5.0625 = 3.395833333 5.0625 + 4.5p + p2 = 3.395833333 Factor a perfect square on the left side: (p + 2.25)(p + 2.25) = 3.395833333 Calculate the square root of the right side: 1.842778699 Break this problem into two subproblems by setting (p + 2.25) equal to 1.842778699 and -1.842778699.Subproblem 1
p + 2.25 = 1.842778699 Simplifying p + 2.25 = 1.842778699 Reorder the terms: 2.25 + p = 1.842778699 Solving 2.25 + p = 1.842778699 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + p = 1.842778699 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + p = 1.842778699 + -2.25 p = 1.842778699 + -2.25 Combine like terms: 1.842778699 + -2.25 = -0.407221301 p = -0.407221301 Simplifying p = -0.407221301Subproblem 2
p + 2.25 = -1.842778699 Simplifying p + 2.25 = -1.842778699 Reorder the terms: 2.25 + p = -1.842778699 Solving 2.25 + p = -1.842778699 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + p = -1.842778699 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + p = -1.842778699 + -2.25 p = -1.842778699 + -2.25 Combine like terms: -1.842778699 + -2.25 = -4.092778699 p = -4.092778699 Simplifying p = -4.092778699Solution
The solution to the problem is based on the solutions from the subproblems. p = {-0.407221301, -4.092778699}
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