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Simplifying 3(3p + 3) * 3p = -3 Reorder the terms: 3(3 + 3p) * 3p = -3 Reorder the terms for easier multiplication: 3 * 3p(3 + 3p) = -3 Multiply 3 * 3 9p(3 + 3p) = -3 (3 * 9p + 3p * 9p) = -3 (27p + 27p2) = -3 Solving 27p + 27p2 = -3 Solving for variable 'p'. Reorder the terms: 3 + 27p + 27p2 = -3 + 3 Combine like terms: -3 + 3 = 0 3 + 27p + 27p2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(1 + 9p + 9p2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(1 + 9p + 9p2)' equal to zero and attempt to solve: Simplifying 1 + 9p + 9p2 = 0 Solving 1 + 9p + 9p2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. 0.1111111111 + p + p2 = 0 Move the constant term to the right: Add '-0.1111111111' to each side of the equation. 0.1111111111 + p + -0.1111111111 + p2 = 0 + -0.1111111111 Reorder the terms: 0.1111111111 + -0.1111111111 + p + p2 = 0 + -0.1111111111 Combine like terms: 0.1111111111 + -0.1111111111 = 0.0000000000 0.0000000000 + p + p2 = 0 + -0.1111111111 p + p2 = 0 + -0.1111111111 Combine like terms: 0 + -0.1111111111 = -0.1111111111 p + p2 = -0.1111111111 The p term is p. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + p2 = -0.1111111111 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + p2 = -0.1111111111 + 0.25 Combine like terms: -0.1111111111 + 0.25 = 0.1388888889 1.25 + p2 = 0.1388888889 Factor a perfect square on the left side: (p + 0.5)(p + 0.5) = 0.1388888889 Calculate the square root of the right side: 0.372677996 Break this problem into two subproblems by setting (p + 0.5) equal to 0.372677996 and -0.372677996.Subproblem 1
p + 0.5 = 0.372677996 Simplifying p + 0.5 = 0.372677996 Reorder the terms: 0.5 + p = 0.372677996 Solving 0.5 + p = 0.372677996 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + p = 0.372677996 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + p = 0.372677996 + -0.5 p = 0.372677996 + -0.5 Combine like terms: 0.372677996 + -0.5 = -0.127322004 p = -0.127322004 Simplifying p = -0.127322004Subproblem 2
p + 0.5 = -0.372677996 Simplifying p + 0.5 = -0.372677996 Reorder the terms: 0.5 + p = -0.372677996 Solving 0.5 + p = -0.372677996 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + p = -0.372677996 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + p = -0.372677996 + -0.5 p = -0.372677996 + -0.5 Combine like terms: -0.372677996 + -0.5 = -0.872677996 p = -0.872677996 Simplifying p = -0.872677996Solution
The solution to the problem is based on the solutions from the subproblems. p = {-0.127322004, -0.872677996}Solution
p = {-0.127322004, -0.872677996}
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