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Simplifying 7p(3p + 4) = -2(2p + -1) + 10 Reorder the terms: 7p(4 + 3p) = -2(2p + -1) + 10 (4 * 7p + 3p * 7p) = -2(2p + -1) + 10 (28p + 21p2) = -2(2p + -1) + 10 Reorder the terms: 28p + 21p2 = -2(-1 + 2p) + 10 28p + 21p2 = (-1 * -2 + 2p * -2) + 10 28p + 21p2 = (2 + -4p) + 10 Reorder the terms: 28p + 21p2 = 2 + 10 + -4p Combine like terms: 2 + 10 = 12 28p + 21p2 = 12 + -4p Solving 28p + 21p2 = 12 + -4p Solving for variable 'p'. Reorder the terms: -12 + 28p + 4p + 21p2 = 12 + -4p + -12 + 4p Combine like terms: 28p + 4p = 32p -12 + 32p + 21p2 = 12 + -4p + -12 + 4p Reorder the terms: -12 + 32p + 21p2 = 12 + -12 + -4p + 4p Combine like terms: 12 + -12 = 0 -12 + 32p + 21p2 = 0 + -4p + 4p -12 + 32p + 21p2 = -4p + 4p Combine like terms: -4p + 4p = 0 -12 + 32p + 21p2 = 0 Begin completing the square. Divide all terms by 21 the coefficient of the squared term: Divide each side by '21'. -0.5714285714 + 1.523809524p + p2 = 0 Move the constant term to the right: Add '0.5714285714' to each side of the equation. -0.5714285714 + 1.523809524p + 0.5714285714 + p2 = 0 + 0.5714285714 Reorder the terms: -0.5714285714 + 0.5714285714 + 1.523809524p + p2 = 0 + 0.5714285714 Combine like terms: -0.5714285714 + 0.5714285714 = 0.0000000000 0.0000000000 + 1.523809524p + p2 = 0 + 0.5714285714 1.523809524p + p2 = 0 + 0.5714285714 Combine like terms: 0 + 0.5714285714 = 0.5714285714 1.523809524p + p2 = 0.5714285714 The p term is 1.523809524p. Take half its coefficient (0.761904762). Square it (0.5804988664) and add it to both sides. Add '0.5804988664' to each side of the equation. 1.523809524p + 0.5804988664 + p2 = 0.5714285714 + 0.5804988664 Reorder the terms: 0.5804988664 + 1.523809524p + p2 = 0.5714285714 + 0.5804988664 Combine like terms: 0.5714285714 + 0.5804988664 = 1.1519274378 0.5804988664 + 1.523809524p + p2 = 1.1519274378 Factor a perfect square on the left side: (p + 0.761904762)(p + 0.761904762) = 1.1519274378 Calculate the square root of the right side: 1.073278826 Break this problem into two subproblems by setting (p + 0.761904762) equal to 1.073278826 and -1.073278826.Subproblem 1
p + 0.761904762 = 1.073278826 Simplifying p + 0.761904762 = 1.073278826 Reorder the terms: 0.761904762 + p = 1.073278826 Solving 0.761904762 + p = 1.073278826 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.761904762' to each side of the equation. 0.761904762 + -0.761904762 + p = 1.073278826 + -0.761904762 Combine like terms: 0.761904762 + -0.761904762 = 0.000000000 0.000000000 + p = 1.073278826 + -0.761904762 p = 1.073278826 + -0.761904762 Combine like terms: 1.073278826 + -0.761904762 = 0.311374064 p = 0.311374064 Simplifying p = 0.311374064Subproblem 2
p + 0.761904762 = -1.073278826 Simplifying p + 0.761904762 = -1.073278826 Reorder the terms: 0.761904762 + p = -1.073278826 Solving 0.761904762 + p = -1.073278826 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.761904762' to each side of the equation. 0.761904762 + -0.761904762 + p = -1.073278826 + -0.761904762 Combine like terms: 0.761904762 + -0.761904762 = 0.000000000 0.000000000 + p = -1.073278826 + -0.761904762 p = -1.073278826 + -0.761904762 Combine like terms: -1.073278826 + -0.761904762 = -1.835183588 p = -1.835183588 Simplifying p = -1.835183588Solution
The solution to the problem is based on the solutions from the subproblems. p = {0.311374064, -1.835183588}
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