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Simplifying 4x2 + 14x + 11 = 0 Reorder the terms: 11 + 14x + 4x2 = 0 Solving 11 + 14x + 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 + 3.5x + x2 = 0 Move the constant term to the right: Add '-2.75' to each side of the equation. 2.75 + 3.5x + -2.75 + x2 = 0 + -2.75 Reorder the terms: 2.75 + -2.75 + 3.5x + x2 = 0 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + 3.5x + x2 = 0 + -2.75 3.5x + x2 = 0 + -2.75 Combine like terms: 0 + -2.75 = -2.75 3.5x + x2 = -2.75 The x term is 3.5x. Take half its coefficient (1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. 3.5x + 3.0625 + x2 = -2.75 + 3.0625 Reorder the terms: 3.0625 + 3.5x + x2 = -2.75 + 3.0625 Combine like terms: -2.75 + 3.0625 = 0.3125 3.0625 + 3.5x + x2 = 0.3125 Factor a perfect square on the left side: (x + 1.75)(x + 1.75) = 0.3125 Calculate the square root of the right side: 0.559016994 Break this problem into two subproblems by setting (x + 1.75) equal to 0.559016994 and -0.559016994.Subproblem 1
x + 1.75 = 0.559016994 Simplifying x + 1.75 = 0.559016994 Reorder the terms: 1.75 + x = 0.559016994 Solving 1.75 + x = 0.559016994 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + x = 0.559016994 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + x = 0.559016994 + -1.75 x = 0.559016994 + -1.75 Combine like terms: 0.559016994 + -1.75 = -1.190983006 x = -1.190983006 Simplifying x = -1.190983006Subproblem 2
x + 1.75 = -0.559016994 Simplifying x + 1.75 = -0.559016994 Reorder the terms: 1.75 + x = -0.559016994 Solving 1.75 + x = -0.559016994 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + x = -0.559016994 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + x = -0.559016994 + -1.75 x = -0.559016994 + -1.75 Combine like terms: -0.559016994 + -1.75 = -2.309016994 x = -2.309016994 Simplifying x = -2.309016994Solution
The solution to the problem is based on the solutions from the subproblems. x = {-1.190983006, -2.309016994}
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