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Simplifying (6p + 3p)(6p + -7) + -1(7p + -4)(p + -2) = 0 Combine like terms: 6p + 3p = 9p (9p)(6p + -7) + -1(7p + -4)(p + -2) = 0 Remove parenthesis around (9p) 9p(6p + -7) + -1(7p + -4)(p + -2) = 0 Reorder the terms: 9p(-7 + 6p) + -1(7p + -4)(p + -2) = 0 (-7 * 9p + 6p * 9p) + -1(7p + -4)(p + -2) = 0 (-63p + 54p2) + -1(7p + -4)(p + -2) = 0 Reorder the terms: -63p + 54p2 + -1(-4 + 7p)(p + -2) = 0 Reorder the terms: -63p + 54p2 + -1(-4 + 7p)(-2 + p) = 0 Multiply (-4 + 7p) * (-2 + p) -63p + 54p2 + -1(-4(-2 + p) + 7p * (-2 + p)) = 0 -63p + 54p2 + -1((-2 * -4 + p * -4) + 7p * (-2 + p)) = 0 -63p + 54p2 + -1((8 + -4p) + 7p * (-2 + p)) = 0 -63p + 54p2 + -1(8 + -4p + (-2 * 7p + p * 7p)) = 0 -63p + 54p2 + -1(8 + -4p + (-14p + 7p2)) = 0 Combine like terms: -4p + -14p = -18p -63p + 54p2 + -1(8 + -18p + 7p2) = 0 -63p + 54p2 + (8 * -1 + -18p * -1 + 7p2 * -1) = 0 -63p + 54p2 + (-8 + 18p + -7p2) = 0 Reorder the terms: -8 + -63p + 18p + 54p2 + -7p2 = 0 Combine like terms: -63p + 18p = -45p -8 + -45p + 54p2 + -7p2 = 0 Combine like terms: 54p2 + -7p2 = 47p2 -8 + -45p + 47p2 = 0 Solving -8 + -45p + 47p2 = 0 Solving for variable 'p'. Begin completing the square. Divide all terms by 47 the coefficient of the squared term: Divide each side by '47'. -0.170212766 + -0.9574468085p + p2 = 0 Move the constant term to the right: Add '0.170212766' to each side of the equation. -0.170212766 + -0.9574468085p + 0.170212766 + p2 = 0 + 0.170212766 Reorder the terms: -0.170212766 + 0.170212766 + -0.9574468085p + p2 = 0 + 0.170212766 Combine like terms: -0.170212766 + 0.170212766 = 0.000000000 0.000000000 + -0.9574468085p + p2 = 0 + 0.170212766 -0.9574468085p + p2 = 0 + 0.170212766 Combine like terms: 0 + 0.170212766 = 0.170212766 -0.9574468085p + p2 = 0.170212766 The p term is -0.9574468085p. Take half its coefficient (-0.4787234043). Square it (0.2291760978) and add it to both sides. Add '0.2291760978' to each side of the equation. -0.9574468085p + 0.2291760978 + p2 = 0.170212766 + 0.2291760978 Reorder the terms: 0.2291760978 + -0.9574468085p + p2 = 0.170212766 + 0.2291760978 Combine like terms: 0.170212766 + 0.2291760978 = 0.3993888638 0.2291760978 + -0.9574468085p + p2 = 0.3993888638 Factor a perfect square on the left side: (p + -0.4787234043)(p + -0.4787234043) = 0.3993888638 Calculate the square root of the right side: 0.631972202 Break this problem into two subproblems by setting (p + -0.4787234043) equal to 0.631972202 and -0.631972202.Subproblem 1
p + -0.4787234043 = 0.631972202 Simplifying p + -0.4787234043 = 0.631972202 Reorder the terms: -0.4787234043 + p = 0.631972202 Solving -0.4787234043 + p = 0.631972202 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.4787234043' to each side of the equation. -0.4787234043 + 0.4787234043 + p = 0.631972202 + 0.4787234043 Combine like terms: -0.4787234043 + 0.4787234043 = 0.0000000000 0.0000000000 + p = 0.631972202 + 0.4787234043 p = 0.631972202 + 0.4787234043 Combine like terms: 0.631972202 + 0.4787234043 = 1.1106956063 p = 1.1106956063 Simplifying p = 1.1106956063Subproblem 2
p + -0.4787234043 = -0.631972202 Simplifying p + -0.4787234043 = -0.631972202 Reorder the terms: -0.4787234043 + p = -0.631972202 Solving -0.4787234043 + p = -0.631972202 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.4787234043' to each side of the equation. -0.4787234043 + 0.4787234043 + p = -0.631972202 + 0.4787234043 Combine like terms: -0.4787234043 + 0.4787234043 = 0.0000000000 0.0000000000 + p = -0.631972202 + 0.4787234043 p = -0.631972202 + 0.4787234043 Combine like terms: -0.631972202 + 0.4787234043 = -0.1532487977 p = -0.1532487977 Simplifying p = -0.1532487977Solution
The solution to the problem is based on the solutions from the subproblems. p = {1.1106956063, -0.1532487977}
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