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Simplifying 3(p2 + -16p + 81) = 81 Reorder the terms: 3(81 + -16p + p2) = 81 (81 * 3 + -16p * 3 + p2 * 3) = 81 (243 + -48p + 3p2) = 81 Solving 243 + -48p + 3p2 = 81 Solving for variable 'p'. Reorder the terms: 243 + -81 + -48p + 3p2 = 81 + -81 Combine like terms: 243 + -81 = 162 162 + -48p + 3p2 = 81 + -81 Combine like terms: 81 + -81 = 0 162 + -48p + 3p2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(54 + -16p + p2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(54 + -16p + p2)' equal to zero and attempt to solve: Simplifying 54 + -16p + p2 = 0 Solving 54 + -16p + p2 = 0 Begin completing the square. Move the constant term to the right: Add '-54' to each side of the equation. 54 + -16p + -54 + p2 = 0 + -54 Reorder the terms: 54 + -54 + -16p + p2 = 0 + -54 Combine like terms: 54 + -54 = 0 0 + -16p + p2 = 0 + -54 -16p + p2 = 0 + -54 Combine like terms: 0 + -54 = -54 -16p + p2 = -54 The p term is -16p. Take half its coefficient (-8). Square it (64) and add it to both sides. Add '64' to each side of the equation. -16p + 64 + p2 = -54 + 64 Reorder the terms: 64 + -16p + p2 = -54 + 64 Combine like terms: -54 + 64 = 10 64 + -16p + p2 = 10 Factor a perfect square on the left side: (p + -8)(p + -8) = 10 Calculate the square root of the right side: 3.16227766 Break this problem into two subproblems by setting (p + -8) equal to 3.16227766 and -3.16227766.Subproblem 1
p + -8 = 3.16227766 Simplifying p + -8 = 3.16227766 Reorder the terms: -8 + p = 3.16227766 Solving -8 + p = 3.16227766 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + p = 3.16227766 + 8 Combine like terms: -8 + 8 = 0 0 + p = 3.16227766 + 8 p = 3.16227766 + 8 Combine like terms: 3.16227766 + 8 = 11.16227766 p = 11.16227766 Simplifying p = 11.16227766Subproblem 2
p + -8 = -3.16227766 Simplifying p + -8 = -3.16227766 Reorder the terms: -8 + p = -3.16227766 Solving -8 + p = -3.16227766 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + p = -3.16227766 + 8 Combine like terms: -8 + 8 = 0 0 + p = -3.16227766 + 8 p = -3.16227766 + 8 Combine like terms: -3.16227766 + 8 = 4.83772234 p = 4.83772234 Simplifying p = 4.83772234Solution
The solution to the problem is based on the solutions from the subproblems. p = {11.16227766, 4.83772234}Solution
p = {11.16227766, 4.83772234}
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