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Simplifying 12x2 + 12x + 1 = 0 Reorder the terms: 1 + 12x + 12x2 = 0 Solving 1 + 12x + 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'. 0.08333333333 + x + x2 = 0 Move the constant term to the right: Add '-0.08333333333' to each side of the equation. 0.08333333333 + x + -0.08333333333 + x2 = 0 + -0.08333333333 Reorder the terms: 0.08333333333 + -0.08333333333 + x + x2 = 0 + -0.08333333333 Combine like terms: 0.08333333333 + -0.08333333333 = 0.00000000000 0.00000000000 + x + x2 = 0 + -0.08333333333 x + x2 = 0 + -0.08333333333 Combine like terms: 0 + -0.08333333333 = -0.08333333333 x + x2 = -0.08333333333 The x term is x. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + x2 = -0.08333333333 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + x2 = -0.08333333333 + 0.25 Combine like terms: -0.08333333333 + 0.25 = 0.16666666667 1.25 + x2 = 0.16666666667 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 0.16666666667 Calculate the square root of the right side: 0.40824829 Break this problem into two subproblems by setting (x + 0.5) equal to 0.40824829 and -0.40824829.Subproblem 1
x + 0.5 = 0.40824829 Simplifying x + 0.5 = 0.40824829 Reorder the terms: 0.5 + x = 0.40824829 Solving 0.5 + x = 0.40824829 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 0.40824829 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 0.40824829 + -0.5 x = 0.40824829 + -0.5 Combine like terms: 0.40824829 + -0.5 = -0.09175171 x = -0.09175171 Simplifying x = -0.09175171Subproblem 2
x + 0.5 = -0.40824829 Simplifying x + 0.5 = -0.40824829 Reorder the terms: 0.5 + x = -0.40824829 Solving 0.5 + x = -0.40824829 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -0.40824829 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -0.40824829 + -0.5 x = -0.40824829 + -0.5 Combine like terms: -0.40824829 + -0.5 = -0.90824829 x = -0.90824829 Simplifying x = -0.90824829Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.09175171, -0.90824829}
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