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Simplifying 7p2 + 6 = 16p Reorder the terms: 6 + 7p2 = 16p Solving 6 + 7p2 = 16p Solving for variable 'p'. Reorder the terms: 6 + -16p + 7p2 = 16p + -16p Combine like terms: 16p + -16p = 0 6 + -16p + 7p2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. 0.8571428571 + -2.285714286p + p2 = 0 Move the constant term to the right: Add '-0.8571428571' to each side of the equation. 0.8571428571 + -2.285714286p + -0.8571428571 + p2 = 0 + -0.8571428571 Reorder the terms: 0.8571428571 + -0.8571428571 + -2.285714286p + p2 = 0 + -0.8571428571 Combine like terms: 0.8571428571 + -0.8571428571 = 0.0000000000 0.0000000000 + -2.285714286p + p2 = 0 + -0.8571428571 -2.285714286p + p2 = 0 + -0.8571428571 Combine like terms: 0 + -0.8571428571 = -0.8571428571 -2.285714286p + p2 = -0.8571428571 The p term is -2.285714286p. Take half its coefficient (-1.142857143). Square it (1.306122449) and add it to both sides. Add '1.306122449' to each side of the equation. -2.285714286p + 1.306122449 + p2 = -0.8571428571 + 1.306122449 Reorder the terms: 1.306122449 + -2.285714286p + p2 = -0.8571428571 + 1.306122449 Combine like terms: -0.8571428571 + 1.306122449 = 0.4489795919 1.306122449 + -2.285714286p + p2 = 0.4489795919 Factor a perfect square on the left side: (p + -1.142857143)(p + -1.142857143) = 0.4489795919 Calculate the square root of the right side: 0.670059394 Break this problem into two subproblems by setting (p + -1.142857143) equal to 0.670059394 and -0.670059394.Subproblem 1
p + -1.142857143 = 0.670059394 Simplifying p + -1.142857143 = 0.670059394 Reorder the terms: -1.142857143 + p = 0.670059394 Solving -1.142857143 + p = 0.670059394 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '1.142857143' to each side of the equation. -1.142857143 + 1.142857143 + p = 0.670059394 + 1.142857143 Combine like terms: -1.142857143 + 1.142857143 = 0.000000000 0.000000000 + p = 0.670059394 + 1.142857143 p = 0.670059394 + 1.142857143 Combine like terms: 0.670059394 + 1.142857143 = 1.812916537 p = 1.812916537 Simplifying p = 1.812916537Subproblem 2
p + -1.142857143 = -0.670059394 Simplifying p + -1.142857143 = -0.670059394 Reorder the terms: -1.142857143 + p = -0.670059394 Solving -1.142857143 + p = -0.670059394 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '1.142857143' to each side of the equation. -1.142857143 + 1.142857143 + p = -0.670059394 + 1.142857143 Combine like terms: -1.142857143 + 1.142857143 = 0.000000000 0.000000000 + p = -0.670059394 + 1.142857143 p = -0.670059394 + 1.142857143 Combine like terms: -0.670059394 + 1.142857143 = 0.472797749 p = 0.472797749 Simplifying p = 0.472797749Solution
The solution to the problem is based on the solutions from the subproblems. p = {1.812916537, 0.472797749}
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