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Simplifying (6p + 3)(6p + -7) + -1(7p + -4)(p + -2) = 0 Reorder the terms: (3 + 6p)(6p + -7) + -1(7p + -4)(p + -2) = 0 Reorder the terms: (3 + 6p)(-7 + 6p) + -1(7p + -4)(p + -2) = 0 Multiply (3 + 6p) * (-7 + 6p) (3(-7 + 6p) + 6p * (-7 + 6p)) + -1(7p + -4)(p + -2) = 0 ((-7 * 3 + 6p * 3) + 6p * (-7 + 6p)) + -1(7p + -4)(p + -2) = 0 ((-21 + 18p) + 6p * (-7 + 6p)) + -1(7p + -4)(p + -2) = 0 (-21 + 18p + (-7 * 6p + 6p * 6p)) + -1(7p + -4)(p + -2) = 0 (-21 + 18p + (-42p + 36p2)) + -1(7p + -4)(p + -2) = 0 Combine like terms: 18p + -42p = -24p (-21 + -24p + 36p2) + -1(7p + -4)(p + -2) = 0 Reorder the terms: -21 + -24p + 36p2 + -1(-4 + 7p)(p + -2) = 0 Reorder the terms: -21 + -24p + 36p2 + -1(-4 + 7p)(-2 + p) = 0 Multiply (-4 + 7p) * (-2 + p) -21 + -24p + 36p2 + -1(-4(-2 + p) + 7p * (-2 + p)) = 0 -21 + -24p + 36p2 + -1((-2 * -4 + p * -4) + 7p * (-2 + p)) = 0 -21 + -24p + 36p2 + -1((8 + -4p) + 7p * (-2 + p)) = 0 -21 + -24p + 36p2 + -1(8 + -4p + (-2 * 7p + p * 7p)) = 0 -21 + -24p + 36p2 + -1(8 + -4p + (-14p + 7p2)) = 0 Combine like terms: -4p + -14p = -18p -21 + -24p + 36p2 + -1(8 + -18p + 7p2) = 0 -21 + -24p + 36p2 + (8 * -1 + -18p * -1 + 7p2 * -1) = 0 -21 + -24p + 36p2 + (-8 + 18p + -7p2) = 0 Reorder the terms: -21 + -8 + -24p + 18p + 36p2 + -7p2 = 0 Combine like terms: -21 + -8 = -29 -29 + -24p + 18p + 36p2 + -7p2 = 0 Combine like terms: -24p + 18p = -6p -29 + -6p + 36p2 + -7p2 = 0 Combine like terms: 36p2 + -7p2 = 29p2 -29 + -6p + 29p2 = 0 Solving -29 + -6p + 29p2 = 0 Solving for variable 'p'. Begin completing the square. Divide all terms by 29 the coefficient of the squared term: Divide each side by '29'. -1 + -0.2068965517p + p2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -0.2068965517p + 1 + p2 = 0 + 1 Reorder the terms: -1 + 1 + -0.2068965517p + p2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -0.2068965517p + p2 = 0 + 1 -0.2068965517p + p2 = 0 + 1 Combine like terms: 0 + 1 = 1 -0.2068965517p + p2 = 1 The p term is -0.2068965517p. Take half its coefficient (-0.1034482759). Square it (0.01070154579) and add it to both sides. Add '0.01070154579' to each side of the equation. -0.2068965517p + 0.01070154579 + p2 = 1 + 0.01070154579 Reorder the terms: 0.01070154579 + -0.2068965517p + p2 = 1 + 0.01070154579 Combine like terms: 1 + 0.01070154579 = 1.01070154579 0.01070154579 + -0.2068965517p + p2 = 1.01070154579 Factor a perfect square on the left side: (p + -0.1034482759)(p + -0.1034482759) = 1.01070154579 Calculate the square root of the right side: 1.005336534 Break this problem into two subproblems by setting (p + -0.1034482759) equal to 1.005336534 and -1.005336534.Subproblem 1
p + -0.1034482759 = 1.005336534 Simplifying p + -0.1034482759 = 1.005336534 Reorder the terms: -0.1034482759 + p = 1.005336534 Solving -0.1034482759 + p = 1.005336534 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.1034482759' to each side of the equation. -0.1034482759 + 0.1034482759 + p = 1.005336534 + 0.1034482759 Combine like terms: -0.1034482759 + 0.1034482759 = 0.0000000000 0.0000000000 + p = 1.005336534 + 0.1034482759 p = 1.005336534 + 0.1034482759 Combine like terms: 1.005336534 + 0.1034482759 = 1.1087848099 p = 1.1087848099 Simplifying p = 1.1087848099Subproblem 2
p + -0.1034482759 = -1.005336534 Simplifying p + -0.1034482759 = -1.005336534 Reorder the terms: -0.1034482759 + p = -1.005336534 Solving -0.1034482759 + p = -1.005336534 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.1034482759' to each side of the equation. -0.1034482759 + 0.1034482759 + p = -1.005336534 + 0.1034482759 Combine like terms: -0.1034482759 + 0.1034482759 = 0.0000000000 0.0000000000 + p = -1.005336534 + 0.1034482759 p = -1.005336534 + 0.1034482759 Combine like terms: -1.005336534 + 0.1034482759 = -0.9018882581 p = -0.9018882581 Simplifying p = -0.9018882581Solution
The solution to the problem is based on the solutions from the subproblems. p = {1.1087848099, -0.9018882581}
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