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Simplifying 9p2 + -4p = 24 + -16p2 Reorder the terms: -4p + 9p2 = 24 + -16p2 Solving -4p + 9p2 = 24 + -16p2 Solving for variable 'p'. Reorder the terms: -24 + -4p + 9p2 + 16p2 = 24 + -16p2 + -24 + 16p2 Combine like terms: 9p2 + 16p2 = 25p2 -24 + -4p + 25p2 = 24 + -16p2 + -24 + 16p2 Reorder the terms: -24 + -4p + 25p2 = 24 + -24 + -16p2 + 16p2 Combine like terms: 24 + -24 = 0 -24 + -4p + 25p2 = 0 + -16p2 + 16p2 -24 + -4p + 25p2 = -16p2 + 16p2 Combine like terms: -16p2 + 16p2 = 0 -24 + -4p + 25p2 = 0 Begin completing the square. Divide all terms by 25 the coefficient of the squared term: Divide each side by '25'. -0.96 + -0.16p + p2 = 0 Move the constant term to the right: Add '0.96' to each side of the equation. -0.96 + -0.16p + 0.96 + p2 = 0 + 0.96 Reorder the terms: -0.96 + 0.96 + -0.16p + p2 = 0 + 0.96 Combine like terms: -0.96 + 0.96 = 0.00 0.00 + -0.16p + p2 = 0 + 0.96 -0.16p + p2 = 0 + 0.96 Combine like terms: 0 + 0.96 = 0.96 -0.16p + p2 = 0.96 The p term is -0.16p. Take half its coefficient (-0.08). Square it (0.0064) and add it to both sides. Add '0.0064' to each side of the equation. -0.16p + 0.0064 + p2 = 0.96 + 0.0064 Reorder the terms: 0.0064 + -0.16p + p2 = 0.96 + 0.0064 Combine like terms: 0.96 + 0.0064 = 0.9664 0.0064 + -0.16p + p2 = 0.9664 Factor a perfect square on the left side: (p + -0.08)(p + -0.08) = 0.9664 Calculate the square root of the right side: 0.983056458 Break this problem into two subproblems by setting (p + -0.08) equal to 0.983056458 and -0.983056458.Subproblem 1
p + -0.08 = 0.983056458 Simplifying p + -0.08 = 0.983056458 Reorder the terms: -0.08 + p = 0.983056458 Solving -0.08 + p = 0.983056458 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.08' to each side of the equation. -0.08 + 0.08 + p = 0.983056458 + 0.08 Combine like terms: -0.08 + 0.08 = 0.00 0.00 + p = 0.983056458 + 0.08 p = 0.983056458 + 0.08 Combine like terms: 0.983056458 + 0.08 = 1.063056458 p = 1.063056458 Simplifying p = 1.063056458Subproblem 2
p + -0.08 = -0.983056458 Simplifying p + -0.08 = -0.983056458 Reorder the terms: -0.08 + p = -0.983056458 Solving -0.08 + p = -0.983056458 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.08' to each side of the equation. -0.08 + 0.08 + p = -0.983056458 + 0.08 Combine like terms: -0.08 + 0.08 = 0.00 0.00 + p = -0.983056458 + 0.08 p = -0.983056458 + 0.08 Combine like terms: -0.983056458 + 0.08 = -0.903056458 p = -0.903056458 Simplifying p = -0.903056458Solution
The solution to the problem is based on the solutions from the subproblems. p = {1.063056458, -0.903056458}
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