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Simplifying 90p3 + -27p2 + -27p = 0 Reorder the terms: -27p + -27p2 + 90p3 = 0 Solving -27p + -27p2 + 90p3 = 0 Solving for variable 'p'. Factor out the Greatest Common Factor (GCF), '9p'. 9p(-3 + -3p + 10p2) = 0 Ignore the factor 9.Subproblem 1
Set the factor 'p' equal to zero and attempt to solve: Simplifying p = 0 Solving p = 0 Move all terms containing p to the left, all other terms to the right. Simplifying p = 0Subproblem 2
Set the factor '(-3 + -3p + 10p2)' equal to zero and attempt to solve: Simplifying -3 + -3p + 10p2 = 0 Solving -3 + -3p + 10p2 = 0 Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. -0.3 + -0.3p + p2 = 0.0 Move the constant term to the right: Add '0.3' to each side of the equation. -0.3 + -0.3p + 0.3 + p2 = 0.0 + 0.3 Reorder the terms: -0.3 + 0.3 + -0.3p + p2 = 0.0 + 0.3 Combine like terms: -0.3 + 0.3 = 0.0 0.0 + -0.3p + p2 = 0.0 + 0.3 -0.3p + p2 = 0.0 + 0.3 Combine like terms: 0.0 + 0.3 = 0.3 -0.3p + p2 = 0.3 The p term is -0.3p. Take half its coefficient (-0.15). Square it (0.0225) and add it to both sides. Add '0.0225' to each side of the equation. -0.3p + 0.0225 + p2 = 0.3 + 0.0225 Reorder the terms: 0.0225 + -0.3p + p2 = 0.3 + 0.0225 Combine like terms: 0.3 + 0.0225 = 0.3225 0.0225 + -0.3p + p2 = 0.3225 Factor a perfect square on the left side: (p + -0.15)(p + -0.15) = 0.3225 Calculate the square root of the right side: 0.567890835 Break this problem into two subproblems by setting (p + -0.15) equal to 0.567890835 and -0.567890835.Subproblem 1
p + -0.15 = 0.567890835 Simplifying p + -0.15 = 0.567890835 Reorder the terms: -0.15 + p = 0.567890835 Solving -0.15 + p = 0.567890835 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.15' to each side of the equation. -0.15 + 0.15 + p = 0.567890835 + 0.15 Combine like terms: -0.15 + 0.15 = 0.00 0.00 + p = 0.567890835 + 0.15 p = 0.567890835 + 0.15 Combine like terms: 0.567890835 + 0.15 = 0.717890835 p = 0.717890835 Simplifying p = 0.717890835Subproblem 2
p + -0.15 = -0.567890835 Simplifying p + -0.15 = -0.567890835 Reorder the terms: -0.15 + p = -0.567890835 Solving -0.15 + p = -0.567890835 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.15' to each side of the equation. -0.15 + 0.15 + p = -0.567890835 + 0.15 Combine like terms: -0.15 + 0.15 = 0.00 0.00 + p = -0.567890835 + 0.15 p = -0.567890835 + 0.15 Combine like terms: -0.567890835 + 0.15 = -0.417890835 p = -0.417890835 Simplifying p = -0.417890835Solution
The solution to the problem is based on the solutions from the subproblems. p = {0.717890835, -0.417890835}Solution
p = {0, 0.717890835, -0.417890835}
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