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Simplifying 5p + 7 + -5p = -7p(p + -3) Reorder the terms: 7 + 5p + -5p = -7p(p + -3) Combine like terms: 5p + -5p = 0 7 + 0 = -7p(p + -3) 7 = -7p(p + -3) Reorder the terms: 7 = -7p(-3 + p) 7 = (-3 * -7p + p * -7p) 7 = (21p + -7p2) Solving 7 = 21p + -7p2 Solving for variable 'p'. Reorder the terms: 7 + -21p + 7p2 = 21p + -21p + -7p2 + 7p2 Combine like terms: 21p + -21p = 0 7 + -21p + 7p2 = 0 + -7p2 + 7p2 7 + -21p + 7p2 = -7p2 + 7p2 Combine like terms: -7p2 + 7p2 = 0 7 + -21p + 7p2 = 0 Factor out the Greatest Common Factor (GCF), '7'. 7(1 + -3p + p2) = 0 Ignore the factor 7.Subproblem 1
Set the factor '(1 + -3p + p2)' equal to zero and attempt to solve: Simplifying 1 + -3p + p2 = 0 Solving 1 + -3p + p2 = 0 Begin completing the square. Move the constant term to the right: Add '-1' to each side of the equation. 1 + -3p + -1 + p2 = 0 + -1 Reorder the terms: 1 + -1 + -3p + p2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -3p + p2 = 0 + -1 -3p + p2 = 0 + -1 Combine like terms: 0 + -1 = -1 -3p + p2 = -1 The p term is -3p. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3p + 2.25 + p2 = -1 + 2.25 Reorder the terms: 2.25 + -3p + p2 = -1 + 2.25 Combine like terms: -1 + 2.25 = 1.25 2.25 + -3p + p2 = 1.25 Factor a perfect square on the left side: (p + -1.5)(p + -1.5) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (p + -1.5) equal to 1.118033989 and -1.118033989.Subproblem 1
p + -1.5 = 1.118033989 Simplifying p + -1.5 = 1.118033989 Reorder the terms: -1.5 + p = 1.118033989 Solving -1.5 + p = 1.118033989 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + p = 1.118033989 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + p = 1.118033989 + 1.5 p = 1.118033989 + 1.5 Combine like terms: 1.118033989 + 1.5 = 2.618033989 p = 2.618033989 Simplifying p = 2.618033989Subproblem 2
p + -1.5 = -1.118033989 Simplifying p + -1.5 = -1.118033989 Reorder the terms: -1.5 + p = -1.118033989 Solving -1.5 + p = -1.118033989 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + p = -1.118033989 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + p = -1.118033989 + 1.5 p = -1.118033989 + 1.5 Combine like terms: -1.118033989 + 1.5 = 0.381966011 p = 0.381966011 Simplifying p = 0.381966011Solution
The solution to the problem is based on the solutions from the subproblems. p = {2.618033989, 0.381966011}Solution
p = {2.618033989, 0.381966011}
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