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Simplifying 8p2 + 3 = 12p Reorder the terms: 3 + 8p2 = 12p Solving 3 + 8p2 = 12p Solving for variable 'p'. Reorder the terms: 3 + -12p + 8p2 = 12p + -12p Combine like terms: 12p + -12p = 0 3 + -12p + 8p2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 0.375 + -1.5p + p2 = 0 Move the constant term to the right: Add '-0.375' to each side of the equation. 0.375 + -1.5p + -0.375 + p2 = 0 + -0.375 Reorder the terms: 0.375 + -0.375 + -1.5p + p2 = 0 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + -1.5p + p2 = 0 + -0.375 -1.5p + p2 = 0 + -0.375 Combine like terms: 0 + -0.375 = -0.375 -1.5p + p2 = -0.375 The p term is -1.5p. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5p + 0.5625 + p2 = -0.375 + 0.5625 Reorder the terms: 0.5625 + -1.5p + p2 = -0.375 + 0.5625 Combine like terms: -0.375 + 0.5625 = 0.1875 0.5625 + -1.5p + p2 = 0.1875 Factor a perfect square on the left side: (p + -0.75)(p + -0.75) = 0.1875 Calculate the square root of the right side: 0.433012702 Break this problem into two subproblems by setting (p + -0.75) equal to 0.433012702 and -0.433012702.Subproblem 1
p + -0.75 = 0.433012702 Simplifying p + -0.75 = 0.433012702 Reorder the terms: -0.75 + p = 0.433012702 Solving -0.75 + p = 0.433012702 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + p = 0.433012702 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + p = 0.433012702 + 0.75 p = 0.433012702 + 0.75 Combine like terms: 0.433012702 + 0.75 = 1.183012702 p = 1.183012702 Simplifying p = 1.183012702Subproblem 2
p + -0.75 = -0.433012702 Simplifying p + -0.75 = -0.433012702 Reorder the terms: -0.75 + p = -0.433012702 Solving -0.75 + p = -0.433012702 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + p = -0.433012702 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + p = -0.433012702 + 0.75 p = -0.433012702 + 0.75 Combine like terms: -0.433012702 + 0.75 = 0.316987298 p = 0.316987298 Simplifying p = 0.316987298Solution
The solution to the problem is based on the solutions from the subproblems. p = {1.183012702, 0.316987298}
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