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Simplifying 4p2 + -8p + -194 = 0 Reorder the terms: -194 + -8p + 4p2 = 0 Solving -194 + -8p + 4p2 = 0 Solving for variable 'p'. Factor out the Greatest Common Factor (GCF), '2'. 2(-97 + -4p + 2p2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-97 + -4p + 2p2)' equal to zero and attempt to solve: Simplifying -97 + -4p + 2p2 = 0 Solving -97 + -4p + 2p2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -48.5 + -2p + p2 = 0 Move the constant term to the right: Add '48.5' to each side of the equation. -48.5 + -2p + 48.5 + p2 = 0 + 48.5 Reorder the terms: -48.5 + 48.5 + -2p + p2 = 0 + 48.5 Combine like terms: -48.5 + 48.5 = 0.0 0.0 + -2p + p2 = 0 + 48.5 -2p + p2 = 0 + 48.5 Combine like terms: 0 + 48.5 = 48.5 -2p + p2 = 48.5 The p term is -2p. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2p + 1 + p2 = 48.5 + 1 Reorder the terms: 1 + -2p + p2 = 48.5 + 1 Combine like terms: 48.5 + 1 = 49.5 1 + -2p + p2 = 49.5 Factor a perfect square on the left side: (p + -1)(p + -1) = 49.5 Calculate the square root of the right side: 7.03562364 Break this problem into two subproblems by setting (p + -1) equal to 7.03562364 and -7.03562364.Subproblem 1
p + -1 = 7.03562364 Simplifying p + -1 = 7.03562364 Reorder the terms: -1 + p = 7.03562364 Solving -1 + p = 7.03562364 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + p = 7.03562364 + 1 Combine like terms: -1 + 1 = 0 0 + p = 7.03562364 + 1 p = 7.03562364 + 1 Combine like terms: 7.03562364 + 1 = 8.03562364 p = 8.03562364 Simplifying p = 8.03562364Subproblem 2
p + -1 = -7.03562364 Simplifying p + -1 = -7.03562364 Reorder the terms: -1 + p = -7.03562364 Solving -1 + p = -7.03562364 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + p = -7.03562364 + 1 Combine like terms: -1 + 1 = 0 0 + p = -7.03562364 + 1 p = -7.03562364 + 1 Combine like terms: -7.03562364 + 1 = -6.03562364 p = -6.03562364 Simplifying p = -6.03562364Solution
The solution to the problem is based on the solutions from the subproblems. p = {8.03562364, -6.03562364}Solution
p = {8.03562364, -6.03562364}
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