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Simplifying 12x2 + 30x + 5 = 0 Reorder the terms: 5 + 30x + 12x2 = 0 Solving 5 + 30x + 12x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. 0.4166666667 + 2.5x + x2 = 0 Move the constant term to the right: Add '-0.4166666667' to each side of the equation. 0.4166666667 + 2.5x + -0.4166666667 + x2 = 0 + -0.4166666667 Reorder the terms: 0.4166666667 + -0.4166666667 + 2.5x + x2 = 0 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + 2.5x + x2 = 0 + -0.4166666667 2.5x + x2 = 0 + -0.4166666667 Combine like terms: 0 + -0.4166666667 = -0.4166666667 2.5x + x2 = -0.4166666667 The x term is 2.5x. Take half its coefficient (1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. 2.5x + 1.5625 + x2 = -0.4166666667 + 1.5625 Reorder the terms: 1.5625 + 2.5x + x2 = -0.4166666667 + 1.5625 Combine like terms: -0.4166666667 + 1.5625 = 1.1458333333 1.5625 + 2.5x + x2 = 1.1458333333 Factor a perfect square on the left side: (x + 1.25)(x + 1.25) = 1.1458333333 Calculate the square root of the right side: 1.070436048 Break this problem into two subproblems by setting (x + 1.25) equal to 1.070436048 and -1.070436048.Subproblem 1
x + 1.25 = 1.070436048 Simplifying x + 1.25 = 1.070436048 Reorder the terms: 1.25 + x = 1.070436048 Solving 1.25 + x = 1.070436048 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + x = 1.070436048 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + x = 1.070436048 + -1.25 x = 1.070436048 + -1.25 Combine like terms: 1.070436048 + -1.25 = -0.179563952 x = -0.179563952 Simplifying x = -0.179563952Subproblem 2
x + 1.25 = -1.070436048 Simplifying x + 1.25 = -1.070436048 Reorder the terms: 1.25 + x = -1.070436048 Solving 1.25 + x = -1.070436048 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + x = -1.070436048 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + x = -1.070436048 + -1.25 x = -1.070436048 + -1.25 Combine like terms: -1.070436048 + -1.25 = -2.320436048 x = -2.320436048 Simplifying x = -2.320436048Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.179563952, -2.320436048}
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