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Simplifying 45x2 + 60x + 11 = 0 Reorder the terms: 11 + 60x + 45x2 = 0 Solving 11 + 60x + 45x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 45 the coefficient of the squared term: Divide each side by '45'. 0.2444444444 + 1.333333333x + x2 = 0 Move the constant term to the right: Add '-0.2444444444' to each side of the equation. 0.2444444444 + 1.333333333x + -0.2444444444 + x2 = 0 + -0.2444444444 Reorder the terms: 0.2444444444 + -0.2444444444 + 1.333333333x + x2 = 0 + -0.2444444444 Combine like terms: 0.2444444444 + -0.2444444444 = 0.0000000000 0.0000000000 + 1.333333333x + x2 = 0 + -0.2444444444 1.333333333x + x2 = 0 + -0.2444444444 Combine like terms: 0 + -0.2444444444 = -0.2444444444 1.333333333x + x2 = -0.2444444444 The x term is 1.333333333x. Take half its coefficient (0.6666666665). Square it (0.4444444442) and add it to both sides. Add '0.4444444442' to each side of the equation. 1.333333333x + 0.4444444442 + x2 = -0.2444444444 + 0.4444444442 Reorder the terms: 0.4444444442 + 1.333333333x + x2 = -0.2444444444 + 0.4444444442 Combine like terms: -0.2444444444 + 0.4444444442 = 0.1999999998 0.4444444442 + 1.333333333x + x2 = 0.1999999998 Factor a perfect square on the left side: (x + 0.6666666665)(x + 0.6666666665) = 0.1999999998 Calculate the square root of the right side: 0.447213595 Break this problem into two subproblems by setting (x + 0.6666666665) equal to 0.447213595 and -0.447213595.Subproblem 1
x + 0.6666666665 = 0.447213595 Simplifying x + 0.6666666665 = 0.447213595 Reorder the terms: 0.6666666665 + x = 0.447213595 Solving 0.6666666665 + x = 0.447213595 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.6666666665' to each side of the equation. 0.6666666665 + -0.6666666665 + x = 0.447213595 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + x = 0.447213595 + -0.6666666665 x = 0.447213595 + -0.6666666665 Combine like terms: 0.447213595 + -0.6666666665 = -0.2194530715 x = -0.2194530715 Simplifying x = -0.2194530715Subproblem 2
x + 0.6666666665 = -0.447213595 Simplifying x + 0.6666666665 = -0.447213595 Reorder the terms: 0.6666666665 + x = -0.447213595 Solving 0.6666666665 + x = -0.447213595 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.6666666665' to each side of the equation. 0.6666666665 + -0.6666666665 + x = -0.447213595 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + x = -0.447213595 + -0.6666666665 x = -0.447213595 + -0.6666666665 Combine like terms: -0.447213595 + -0.6666666665 = -1.1138802615 x = -1.1138802615 Simplifying x = -1.1138802615Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.2194530715, -1.1138802615}
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