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Simplifying -5p(7p + 4) + -1(p + 1) = 0 Reorder the terms: -5p(4 + 7p) + -1(p + 1) = 0 (4 * -5p + 7p * -5p) + -1(p + 1) = 0 (-20p + -35p2) + -1(p + 1) = 0 Reorder the terms: -20p + -35p2 + -1(1 + p) = 0 -20p + -35p2 + (1 * -1 + p * -1) = 0 -20p + -35p2 + (-1 + -1p) = 0 Reorder the terms: -1 + -20p + -1p + -35p2 = 0 Combine like terms: -20p + -1p = -21p -1 + -21p + -35p2 = 0 Solving -1 + -21p + -35p2 = 0 Solving for variable 'p'. Factor out the Greatest Common Factor (GCF), '-1'. -1(1 + 21p + 35p2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(1 + 21p + 35p2)' equal to zero and attempt to solve: Simplifying 1 + 21p + 35p2 = 0 Solving 1 + 21p + 35p2 = 0 Begin completing the square. Divide all terms by 35 the coefficient of the squared term: Divide each side by '35'. 0.02857142857 + 0.6p + p2 = 0 Move the constant term to the right: Add '-0.02857142857' to each side of the equation. 0.02857142857 + 0.6p + -0.02857142857 + p2 = 0 + -0.02857142857 Reorder the terms: 0.02857142857 + -0.02857142857 + 0.6p + p2 = 0 + -0.02857142857 Combine like terms: 0.02857142857 + -0.02857142857 = 0.00000000000 0.00000000000 + 0.6p + p2 = 0 + -0.02857142857 0.6p + p2 = 0 + -0.02857142857 Combine like terms: 0 + -0.02857142857 = -0.02857142857 0.6p + p2 = -0.02857142857 The p term is 0.6p. Take half its coefficient (0.3). Square it (0.09) and add it to both sides. Add '0.09' to each side of the equation. 0.6p + 0.09 + p2 = -0.02857142857 + 0.09 Reorder the terms: 0.09 + 0.6p + p2 = -0.02857142857 + 0.09 Combine like terms: -0.02857142857 + 0.09 = 0.06142857143 0.09 + 0.6p + p2 = 0.06142857143 Factor a perfect square on the left side: (p + 0.3)(p + 0.3) = 0.06142857143 Calculate the square root of the right side: 0.24784788 Break this problem into two subproblems by setting (p + 0.3) equal to 0.24784788 and -0.24784788.Subproblem 1
p + 0.3 = 0.24784788 Simplifying p + 0.3 = 0.24784788 Reorder the terms: 0.3 + p = 0.24784788 Solving 0.3 + p = 0.24784788 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.3' to each side of the equation. 0.3 + -0.3 + p = 0.24784788 + -0.3 Combine like terms: 0.3 + -0.3 = 0.0 0.0 + p = 0.24784788 + -0.3 p = 0.24784788 + -0.3 Combine like terms: 0.24784788 + -0.3 = -0.05215212 p = -0.05215212 Simplifying p = -0.05215212Subproblem 2
p + 0.3 = -0.24784788 Simplifying p + 0.3 = -0.24784788 Reorder the terms: 0.3 + p = -0.24784788 Solving 0.3 + p = -0.24784788 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.3' to each side of the equation. 0.3 + -0.3 + p = -0.24784788 + -0.3 Combine like terms: 0.3 + -0.3 = 0.0 0.0 + p = -0.24784788 + -0.3 p = -0.24784788 + -0.3 Combine like terms: -0.24784788 + -0.3 = -0.54784788 p = -0.54784788 Simplifying p = -0.54784788Solution
The solution to the problem is based on the solutions from the subproblems. p = {-0.05215212, -0.54784788}Solution
p = {-0.05215212, -0.54784788}
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