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Simplifying p2 + 2p * 0.583 + 0.34 = 1 Reorder the terms for easier multiplication: p2 + 2 * 0.583p + 0.34 = 1 Multiply 2 * 0.583 p2 + 1.166p + 0.34 = 1 Reorder the terms: 0.34 + 1.166p + p2 = 1 Solving 0.34 + 1.166p + p2 = 1 Solving for variable 'p'. Reorder the terms: 0.34 + -1 + 1.166p + p2 = 1 + -1 Combine like terms: 0.34 + -1 = -0.66 -0.66 + 1.166p + p2 = 1 + -1 Combine like terms: 1 + -1 = 0 -0.66 + 1.166p + p2 = 0 Begin completing the square. Move the constant term to the right: Add '0.66' to each side of the equation. -0.66 + 1.166p + 0.66 + p2 = 0 + 0.66 Reorder the terms: -0.66 + 0.66 + 1.166p + p2 = 0 + 0.66 Combine like terms: -0.66 + 0.66 = 0.00 0.00 + 1.166p + p2 = 0 + 0.66 1.166p + p2 = 0 + 0.66 Combine like terms: 0 + 0.66 = 0.66 1.166p + p2 = 0.66 The p term is 1.166p. Take half its coefficient (0.583). Square it (0.339889) and add it to both sides. Add '0.339889' to each side of the equation. 1.166p + 0.339889 + p2 = 0.66 + 0.339889 Reorder the terms: 0.339889 + 1.166p + p2 = 0.66 + 0.339889 Combine like terms: 0.66 + 0.339889 = 0.999889 0.339889 + 1.166p + p2 = 0.999889 Factor a perfect square on the left side: (p + 0.583)(p + 0.583) = 0.999889 Calculate the square root of the right side: 0.999944498 Break this problem into two subproblems by setting (p + 0.583) equal to 0.999944498 and -0.999944498.Subproblem 1
p + 0.583 = 0.999944498 Simplifying p + 0.583 = 0.999944498 Reorder the terms: 0.583 + p = 0.999944498 Solving 0.583 + p = 0.999944498 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.583' to each side of the equation. 0.583 + -0.583 + p = 0.999944498 + -0.583 Combine like terms: 0.583 + -0.583 = 0.000 0.000 + p = 0.999944498 + -0.583 p = 0.999944498 + -0.583 Combine like terms: 0.999944498 + -0.583 = 0.416944498 p = 0.416944498 Simplifying p = 0.416944498Subproblem 2
p + 0.583 = -0.999944498 Simplifying p + 0.583 = -0.999944498 Reorder the terms: 0.583 + p = -0.999944498 Solving 0.583 + p = -0.999944498 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.583' to each side of the equation. 0.583 + -0.583 + p = -0.999944498 + -0.583 Combine like terms: 0.583 + -0.583 = 0.000 0.000 + p = -0.999944498 + -0.583 p = -0.999944498 + -0.583 Combine like terms: -0.999944498 + -0.583 = -1.582944498 p = -1.582944498 Simplifying p = -1.582944498Solution
The solution to the problem is based on the solutions from the subproblems. p = {0.416944498, -1.582944498}
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