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Simplifying 24x2 + 12x + 1 = 0 Reorder the terms: 1 + 12x + 24x2 = 0 Solving 1 + 12x + 24x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 24 the coefficient of the squared term: Divide each side by '24'. 0.04166666667 + 0.5x + x2 = 0 Move the constant term to the right: Add '-0.04166666667' to each side of the equation. 0.04166666667 + 0.5x + -0.04166666667 + x2 = 0 + -0.04166666667 Reorder the terms: 0.04166666667 + -0.04166666667 + 0.5x + x2 = 0 + -0.04166666667 Combine like terms: 0.04166666667 + -0.04166666667 = 0.00000000000 0.00000000000 + 0.5x + x2 = 0 + -0.04166666667 0.5x + x2 = 0 + -0.04166666667 Combine like terms: 0 + -0.04166666667 = -0.04166666667 0.5x + x2 = -0.04166666667 The x term is 0.5x. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5x + 0.0625 + x2 = -0.04166666667 + 0.0625 Reorder the terms: 0.0625 + 0.5x + x2 = -0.04166666667 + 0.0625 Combine like terms: -0.04166666667 + 0.0625 = 0.02083333333 0.0625 + 0.5x + x2 = 0.02083333333 Factor a perfect square on the left side: (x + 0.25)(x + 0.25) = 0.02083333333 Calculate the square root of the right side: 0.144337567 Break this problem into two subproblems by setting (x + 0.25) equal to 0.144337567 and -0.144337567.Subproblem 1
x + 0.25 = 0.144337567 Simplifying x + 0.25 = 0.144337567 Reorder the terms: 0.25 + x = 0.144337567 Solving 0.25 + x = 0.144337567 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = 0.144337567 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 0.144337567 + -0.25 x = 0.144337567 + -0.25 Combine like terms: 0.144337567 + -0.25 = -0.105662433 x = -0.105662433 Simplifying x = -0.105662433Subproblem 2
x + 0.25 = -0.144337567 Simplifying x + 0.25 = -0.144337567 Reorder the terms: 0.25 + x = -0.144337567 Solving 0.25 + x = -0.144337567 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = -0.144337567 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -0.144337567 + -0.25 x = -0.144337567 + -0.25 Combine like terms: -0.144337567 + -0.25 = -0.394337567 x = -0.394337567 Simplifying x = -0.394337567Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.105662433, -0.394337567}
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