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Simplifying 6p2 + 5 = 12p Reorder the terms: 5 + 6p2 = 12p Solving 5 + 6p2 = 12p Solving for variable 'p'. Reorder the terms: 5 + -12p + 6p2 = 12p + -12p Combine like terms: 12p + -12p = 0 5 + -12p + 6p2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. 0.8333333333 + -2p + p2 = 0 Move the constant term to the right: Add '-0.8333333333' to each side of the equation. 0.8333333333 + -2p + -0.8333333333 + p2 = 0 + -0.8333333333 Reorder the terms: 0.8333333333 + -0.8333333333 + -2p + p2 = 0 + -0.8333333333 Combine like terms: 0.8333333333 + -0.8333333333 = 0.0000000000 0.0000000000 + -2p + p2 = 0 + -0.8333333333 -2p + p2 = 0 + -0.8333333333 Combine like terms: 0 + -0.8333333333 = -0.8333333333 -2p + p2 = -0.8333333333 The p term is -2p. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2p + 1 + p2 = -0.8333333333 + 1 Reorder the terms: 1 + -2p + p2 = -0.8333333333 + 1 Combine like terms: -0.8333333333 + 1 = 0.1666666667 1 + -2p + p2 = 0.1666666667 Factor a perfect square on the left side: (p + -1)(p + -1) = 0.1666666667 Calculate the square root of the right side: 0.408248291 Break this problem into two subproblems by setting (p + -1) equal to 0.408248291 and -0.408248291.Subproblem 1
p + -1 = 0.408248291 Simplifying p + -1 = 0.408248291 Reorder the terms: -1 + p = 0.408248291 Solving -1 + p = 0.408248291 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + p = 0.408248291 + 1 Combine like terms: -1 + 1 = 0 0 + p = 0.408248291 + 1 p = 0.408248291 + 1 Combine like terms: 0.408248291 + 1 = 1.408248291 p = 1.408248291 Simplifying p = 1.408248291Subproblem 2
p + -1 = -0.408248291 Simplifying p + -1 = -0.408248291 Reorder the terms: -1 + p = -0.408248291 Solving -1 + p = -0.408248291 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + p = -0.408248291 + 1 Combine like terms: -1 + 1 = 0 0 + p = -0.408248291 + 1 p = -0.408248291 + 1 Combine like terms: -0.408248291 + 1 = 0.591751709 p = 0.591751709 Simplifying p = 0.591751709Solution
The solution to the problem is based on the solutions from the subproblems. p = {1.408248291, 0.591751709}
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