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Simplifying 8 = p(12p + -3) Reorder the terms: 8 = p(-3 + 12p) 8 = (-3 * p + 12p * p) 8 = (-3p + 12p2) Solving 8 = -3p + 12p2 Solving for variable 'p'. Reorder the terms: 8 + 3p + -12p2 = -3p + 3p + 12p2 + -12p2 Combine like terms: -3p + 3p = 0 8 + 3p + -12p2 = 0 + 12p2 + -12p2 8 + 3p + -12p2 = 12p2 + -12p2 Combine like terms: 12p2 + -12p2 = 0 8 + 3p + -12p2 = 0 Begin completing the square. Divide all terms by -12 the coefficient of the squared term: Divide each side by '-12'. -0.6666666667 + -0.25p + p2 = 0 Move the constant term to the right: Add '0.6666666667' to each side of the equation. -0.6666666667 + -0.25p + 0.6666666667 + p2 = 0 + 0.6666666667 Reorder the terms: -0.6666666667 + 0.6666666667 + -0.25p + p2 = 0 + 0.6666666667 Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000 0.0000000000 + -0.25p + p2 = 0 + 0.6666666667 -0.25p + p2 = 0 + 0.6666666667 Combine like terms: 0 + 0.6666666667 = 0.6666666667 -0.25p + p2 = 0.6666666667 The p term is -0.25p. Take half its coefficient (-0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. -0.25p + 0.015625 + p2 = 0.6666666667 + 0.015625 Reorder the terms: 0.015625 + -0.25p + p2 = 0.6666666667 + 0.015625 Combine like terms: 0.6666666667 + 0.015625 = 0.6822916667 0.015625 + -0.25p + p2 = 0.6822916667 Factor a perfect square on the left side: (p + -0.125)(p + -0.125) = 0.6822916667 Calculate the square root of the right side: 0.826009483 Break this problem into two subproblems by setting (p + -0.125) equal to 0.826009483 and -0.826009483.Subproblem 1
p + -0.125 = 0.826009483 Simplifying p + -0.125 = 0.826009483 Reorder the terms: -0.125 + p = 0.826009483 Solving -0.125 + p = 0.826009483 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + p = 0.826009483 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + p = 0.826009483 + 0.125 p = 0.826009483 + 0.125 Combine like terms: 0.826009483 + 0.125 = 0.951009483 p = 0.951009483 Simplifying p = 0.951009483Subproblem 2
p + -0.125 = -0.826009483 Simplifying p + -0.125 = -0.826009483 Reorder the terms: -0.125 + p = -0.826009483 Solving -0.125 + p = -0.826009483 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + p = -0.826009483 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + p = -0.826009483 + 0.125 p = -0.826009483 + 0.125 Combine like terms: -0.826009483 + 0.125 = -0.701009483 p = -0.701009483 Simplifying p = -0.701009483Solution
The solution to the problem is based on the solutions from the subproblems. p = {0.951009483, -0.701009483}
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