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Simplifying 3p * 3p + 11p + -4 = 0 Reorder the terms for easier multiplication: 3 * 3p * p + 11p + -4 = 0 Multiply 3 * 3 9p * p + 11p + -4 = 0 Multiply p * p 9p2 + 11p + -4 = 0 Reorder the terms: -4 + 11p + 9p2 = 0 Solving -4 + 11p + 9p2 = 0 Solving for variable 'p'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -0.4444444444 + 1.222222222p + p2 = 0 Move the constant term to the right: Add '0.4444444444' to each side of the equation. -0.4444444444 + 1.222222222p + 0.4444444444 + p2 = 0 + 0.4444444444 Reorder the terms: -0.4444444444 + 0.4444444444 + 1.222222222p + p2 = 0 + 0.4444444444 Combine like terms: -0.4444444444 + 0.4444444444 = 0.0000000000 0.0000000000 + 1.222222222p + p2 = 0 + 0.4444444444 1.222222222p + p2 = 0 + 0.4444444444 Combine like terms: 0 + 0.4444444444 = 0.4444444444 1.222222222p + p2 = 0.4444444444 The p term is 1.222222222p. Take half its coefficient (0.611111111). Square it (0.3734567900) and add it to both sides. Add '0.3734567900' to each side of the equation. 1.222222222p + 0.3734567900 + p2 = 0.4444444444 + 0.3734567900 Reorder the terms: 0.3734567900 + 1.222222222p + p2 = 0.4444444444 + 0.3734567900 Combine like terms: 0.4444444444 + 0.3734567900 = 0.8179012344 0.3734567900 + 1.222222222p + p2 = 0.8179012344 Factor a perfect square on the left side: (p + 0.611111111)(p + 0.611111111) = 0.8179012344 Calculate the square root of the right side: 0.904378922 Break this problem into two subproblems by setting (p + 0.611111111) equal to 0.904378922 and -0.904378922.Subproblem 1
p + 0.611111111 = 0.904378922 Simplifying p + 0.611111111 = 0.904378922 Reorder the terms: 0.611111111 + p = 0.904378922 Solving 0.611111111 + p = 0.904378922 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.611111111' to each side of the equation. 0.611111111 + -0.611111111 + p = 0.904378922 + -0.611111111 Combine like terms: 0.611111111 + -0.611111111 = 0.000000000 0.000000000 + p = 0.904378922 + -0.611111111 p = 0.904378922 + -0.611111111 Combine like terms: 0.904378922 + -0.611111111 = 0.293267811 p = 0.293267811 Simplifying p = 0.293267811Subproblem 2
p + 0.611111111 = -0.904378922 Simplifying p + 0.611111111 = -0.904378922 Reorder the terms: 0.611111111 + p = -0.904378922 Solving 0.611111111 + p = -0.904378922 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.611111111' to each side of the equation. 0.611111111 + -0.611111111 + p = -0.904378922 + -0.611111111 Combine like terms: 0.611111111 + -0.611111111 = 0.000000000 0.000000000 + p = -0.904378922 + -0.611111111 p = -0.904378922 + -0.611111111 Combine like terms: -0.904378922 + -0.611111111 = -1.515490033 p = -1.515490033 Simplifying p = -1.515490033Solution
The solution to the problem is based on the solutions from the subproblems. p = {0.293267811, -1.515490033}
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