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Simplifying 5x2 + 15x + 11 = 0 Reorder the terms: 11 + 15x + 5x2 = 0 Solving 11 + 15x + 5x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 2.2 + 3x + x2 = 0 Move the constant term to the right: Add '-2.2' to each side of the equation. 2.2 + 3x + -2.2 + x2 = 0 + -2.2 Reorder the terms: 2.2 + -2.2 + 3x + x2 = 0 + -2.2 Combine like terms: 2.2 + -2.2 = 0.0 0.0 + 3x + x2 = 0 + -2.2 3x + x2 = 0 + -2.2 Combine like terms: 0 + -2.2 = -2.2 3x + x2 = -2.2 The x term is 3x. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3x + 2.25 + x2 = -2.2 + 2.25 Reorder the terms: 2.25 + 3x + x2 = -2.2 + 2.25 Combine like terms: -2.2 + 2.25 = 0.05 2.25 + 3x + x2 = 0.05 Factor a perfect square on the left side: (x + 1.5)(x + 1.5) = 0.05 Calculate the square root of the right side: 0.223606798 Break this problem into two subproblems by setting (x + 1.5) equal to 0.223606798 and -0.223606798.Subproblem 1
x + 1.5 = 0.223606798 Simplifying x + 1.5 = 0.223606798 Reorder the terms: 1.5 + x = 0.223606798 Solving 1.5 + x = 0.223606798 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = 0.223606798 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 0.223606798 + -1.5 x = 0.223606798 + -1.5 Combine like terms: 0.223606798 + -1.5 = -1.276393202 x = -1.276393202 Simplifying x = -1.276393202Subproblem 2
x + 1.5 = -0.223606798 Simplifying x + 1.5 = -0.223606798 Reorder the terms: 1.5 + x = -0.223606798 Solving 1.5 + x = -0.223606798 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = -0.223606798 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -0.223606798 + -1.5 x = -0.223606798 + -1.5 Combine like terms: -0.223606798 + -1.5 = -1.723606798 x = -1.723606798 Simplifying x = -1.723606798Solution
The solution to the problem is based on the solutions from the subproblems. x = {-1.276393202, -1.723606798}
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