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Simplifying 7p2 + 31p + 20 = 0 Reorder the terms: 20 + 31p + 7p2 = 0 Solving 20 + 31p + 7p2 = 0 Solving for variable 'p'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. 2.857142857 + 4.428571429p + p2 = 0 Move the constant term to the right: Add '-2.857142857' to each side of the equation. 2.857142857 + 4.428571429p + -2.857142857 + p2 = 0 + -2.857142857 Reorder the terms: 2.857142857 + -2.857142857 + 4.428571429p + p2 = 0 + -2.857142857 Combine like terms: 2.857142857 + -2.857142857 = 0.000000000 0.000000000 + 4.428571429p + p2 = 0 + -2.857142857 4.428571429p + p2 = 0 + -2.857142857 Combine like terms: 0 + -2.857142857 = -2.857142857 4.428571429p + p2 = -2.857142857 The p term is 4.428571429p. Take half its coefficient (2.214285715). Square it (4.903061228) and add it to both sides. Add '4.903061228' to each side of the equation. 4.428571429p + 4.903061228 + p2 = -2.857142857 + 4.903061228 Reorder the terms: 4.903061228 + 4.428571429p + p2 = -2.857142857 + 4.903061228 Combine like terms: -2.857142857 + 4.903061228 = 2.045918371 4.903061228 + 4.428571429p + p2 = 2.045918371 Factor a perfect square on the left side: (p + 2.214285715)(p + 2.214285715) = 2.045918371 Calculate the square root of the right side: 1.430356029 Break this problem into two subproblems by setting (p + 2.214285715) equal to 1.430356029 and -1.430356029.Subproblem 1
p + 2.214285715 = 1.430356029 Simplifying p + 2.214285715 = 1.430356029 Reorder the terms: 2.214285715 + p = 1.430356029 Solving 2.214285715 + p = 1.430356029 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-2.214285715' to each side of the equation. 2.214285715 + -2.214285715 + p = 1.430356029 + -2.214285715 Combine like terms: 2.214285715 + -2.214285715 = 0.000000000 0.000000000 + p = 1.430356029 + -2.214285715 p = 1.430356029 + -2.214285715 Combine like terms: 1.430356029 + -2.214285715 = -0.783929686 p = -0.783929686 Simplifying p = -0.783929686Subproblem 2
p + 2.214285715 = -1.430356029 Simplifying p + 2.214285715 = -1.430356029 Reorder the terms: 2.214285715 + p = -1.430356029 Solving 2.214285715 + p = -1.430356029 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-2.214285715' to each side of the equation. 2.214285715 + -2.214285715 + p = -1.430356029 + -2.214285715 Combine like terms: 2.214285715 + -2.214285715 = 0.000000000 0.000000000 + p = -1.430356029 + -2.214285715 p = -1.430356029 + -2.214285715 Combine like terms: -1.430356029 + -2.214285715 = -3.644641744 p = -3.644641744 Simplifying p = -3.644641744Solution
The solution to the problem is based on the solutions from the subproblems. p = {-0.783929686, -3.644641744}
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