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Simplifying 6p2 + 4 = 12p Reorder the terms: 4 + 6p2 = 12p Solving 4 + 6p2 = 12p Solving for variable 'p'. Reorder the terms: 4 + -12p + 6p2 = 12p + -12p Combine like terms: 12p + -12p = 0 4 + -12p + 6p2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(2 + -6p + 3p2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(2 + -6p + 3p2)' equal to zero and attempt to solve: Simplifying 2 + -6p + 3p2 = 0 Solving 2 + -6p + 3p2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 0.6666666667 + -2p + p2 = 0 Move the constant term to the right: Add '-0.6666666667' to each side of the equation. 0.6666666667 + -2p + -0.6666666667 + p2 = 0 + -0.6666666667 Reorder the terms: 0.6666666667 + -0.6666666667 + -2p + p2 = 0 + -0.6666666667 Combine like terms: 0.6666666667 + -0.6666666667 = 0.0000000000 0.0000000000 + -2p + p2 = 0 + -0.6666666667 -2p + p2 = 0 + -0.6666666667 Combine like terms: 0 + -0.6666666667 = -0.6666666667 -2p + p2 = -0.6666666667 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 = -0.6666666667 + 1 Reorder the terms: 1 + -2p + p2 = -0.6666666667 + 1 Combine like terms: -0.6666666667 + 1 = 0.3333333333 1 + -2p + p2 = 0.3333333333 Factor a perfect square on the left side: (p + -1)(p + -1) = 0.3333333333 Calculate the square root of the right side: 0.577350269 Break this problem into two subproblems by setting (p + -1) equal to 0.577350269 and -0.577350269.Subproblem 1
p + -1 = 0.577350269 Simplifying p + -1 = 0.577350269 Reorder the terms: -1 + p = 0.577350269 Solving -1 + p = 0.577350269 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 = 0.577350269 + 1 Combine like terms: -1 + 1 = 0 0 + p = 0.577350269 + 1 p = 0.577350269 + 1 Combine like terms: 0.577350269 + 1 = 1.577350269 p = 1.577350269 Simplifying p = 1.577350269Subproblem 2
p + -1 = -0.577350269 Simplifying p + -1 = -0.577350269 Reorder the terms: -1 + p = -0.577350269 Solving -1 + p = -0.577350269 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 = -0.577350269 + 1 Combine like terms: -1 + 1 = 0 0 + p = -0.577350269 + 1 p = -0.577350269 + 1 Combine like terms: -0.577350269 + 1 = 0.422649731 p = 0.422649731 Simplifying p = 0.422649731Solution
The solution to the problem is based on the solutions from the subproblems. p = {1.577350269, 0.422649731}Solution
p = {1.577350269, 0.422649731}
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