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Simplifying (6p2 + 4p + 5) + -1(2p2 + -5p + 1) = 0 Reorder the terms: (5 + 4p + 6p2) + -1(2p2 + -5p + 1) = 0 Remove parenthesis around (5 + 4p + 6p2) 5 + 4p + 6p2 + -1(2p2 + -5p + 1) = 0 Reorder the terms: 5 + 4p + 6p2 + -1(1 + -5p + 2p2) = 0 5 + 4p + 6p2 + (1 * -1 + -5p * -1 + 2p2 * -1) = 0 5 + 4p + 6p2 + (-1 + 5p + -2p2) = 0 Reorder the terms: 5 + -1 + 4p + 5p + 6p2 + -2p2 = 0 Combine like terms: 5 + -1 = 4 4 + 4p + 5p + 6p2 + -2p2 = 0 Combine like terms: 4p + 5p = 9p 4 + 9p + 6p2 + -2p2 = 0 Combine like terms: 6p2 + -2p2 = 4p2 4 + 9p + 4p2 = 0 Solving 4 + 9p + 4p2 = 0 Solving for variable 'p'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 1 + 2.25p + p2 = 0 Move the constant term to the right: Add '-1' to each side of the equation. 1 + 2.25p + -1 + p2 = 0 + -1 Reorder the terms: 1 + -1 + 2.25p + p2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + 2.25p + p2 = 0 + -1 2.25p + p2 = 0 + -1 Combine like terms: 0 + -1 = -1 2.25p + p2 = -1 The p term is 2.25p. Take half its coefficient (1.125). Square it (1.265625) and add it to both sides. Add '1.265625' to each side of the equation. 2.25p + 1.265625 + p2 = -1 + 1.265625 Reorder the terms: 1.265625 + 2.25p + p2 = -1 + 1.265625 Combine like terms: -1 + 1.265625 = 0.265625 1.265625 + 2.25p + p2 = 0.265625 Factor a perfect square on the left side: (p + 1.125)(p + 1.125) = 0.265625 Calculate the square root of the right side: 0.515388203 Break this problem into two subproblems by setting (p + 1.125) equal to 0.515388203 and -0.515388203.Subproblem 1
p + 1.125 = 0.515388203 Simplifying p + 1.125 = 0.515388203 Reorder the terms: 1.125 + p = 0.515388203 Solving 1.125 + p = 0.515388203 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-1.125' to each side of the equation. 1.125 + -1.125 + p = 0.515388203 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + p = 0.515388203 + -1.125 p = 0.515388203 + -1.125 Combine like terms: 0.515388203 + -1.125 = -0.609611797 p = -0.609611797 Simplifying p = -0.609611797Subproblem 2
p + 1.125 = -0.515388203 Simplifying p + 1.125 = -0.515388203 Reorder the terms: 1.125 + p = -0.515388203 Solving 1.125 + p = -0.515388203 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-1.125' to each side of the equation. 1.125 + -1.125 + p = -0.515388203 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + p = -0.515388203 + -1.125 p = -0.515388203 + -1.125 Combine like terms: -0.515388203 + -1.125 = -1.640388203 p = -1.640388203 Simplifying p = -1.640388203Solution
The solution to the problem is based on the solutions from the subproblems. p = {-0.609611797, -1.640388203}
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