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Simplifying 4x2 + 13.5x + 11 = 0 Reorder the terms: 11 + 13.5x + 4x2 = 0 Solving 11 + 13.5x + 4x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 2.75 + 3.375x + x2 = 0 Move the constant term to the right: Add '-2.75' to each side of the equation. 2.75 + 3.375x + -2.75 + x2 = 0 + -2.75 Reorder the terms: 2.75 + -2.75 + 3.375x + x2 = 0 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + 3.375x + x2 = 0 + -2.75 3.375x + x2 = 0 + -2.75 Combine like terms: 0 + -2.75 = -2.75 3.375x + x2 = -2.75 The x term is 3.375x. Take half its coefficient (1.6875). Square it (2.84765625) and add it to both sides. Add '2.84765625' to each side of the equation. 3.375x + 2.84765625 + x2 = -2.75 + 2.84765625 Reorder the terms: 2.84765625 + 3.375x + x2 = -2.75 + 2.84765625 Combine like terms: -2.75 + 2.84765625 = 0.09765625 2.84765625 + 3.375x + x2 = 0.09765625 Factor a perfect square on the left side: (x + 1.6875)(x + 1.6875) = 0.09765625 Calculate the square root of the right side: 0.3125 Break this problem into two subproblems by setting (x + 1.6875) equal to 0.3125 and -0.3125.Subproblem 1
x + 1.6875 = 0.3125 Simplifying x + 1.6875 = 0.3125 Reorder the terms: 1.6875 + x = 0.3125 Solving 1.6875 + x = 0.3125 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.6875' to each side of the equation. 1.6875 + -1.6875 + x = 0.3125 + -1.6875 Combine like terms: 1.6875 + -1.6875 = 0.0000 0.0000 + x = 0.3125 + -1.6875 x = 0.3125 + -1.6875 Combine like terms: 0.3125 + -1.6875 = -1.375 x = -1.375 Simplifying x = -1.375Subproblem 2
x + 1.6875 = -0.3125 Simplifying x + 1.6875 = -0.3125 Reorder the terms: 1.6875 + x = -0.3125 Solving 1.6875 + x = -0.3125 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.6875' to each side of the equation. 1.6875 + -1.6875 + x = -0.3125 + -1.6875 Combine like terms: 1.6875 + -1.6875 = 0.0000 0.0000 + x = -0.3125 + -1.6875 x = -0.3125 + -1.6875 Combine like terms: -0.3125 + -1.6875 = -2 x = -2 Simplifying x = -2Solution
The solution to the problem is based on the solutions from the subproblems. x = {-1.375, -2}
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