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Simplifying 1 + 5x = 3 + -4x(2 + 3x) 1 + 5x = 3 + (2 * -4x + 3x * -4x) 1 + 5x = 3 + (-8x + -12x2) Solving 1 + 5x = 3 + -8x + -12x2 Solving for variable 'x'. Reorder the terms: 1 + -3 + 5x + 8x + 12x2 = 3 + -8x + -12x2 + -3 + 8x + 12x2 Combine like terms: 1 + -3 = -2 -2 + 5x + 8x + 12x2 = 3 + -8x + -12x2 + -3 + 8x + 12x2 Combine like terms: 5x + 8x = 13x -2 + 13x + 12x2 = 3 + -8x + -12x2 + -3 + 8x + 12x2 Reorder the terms: -2 + 13x + 12x2 = 3 + -3 + -8x + 8x + -12x2 + 12x2 Combine like terms: 3 + -3 = 0 -2 + 13x + 12x2 = 0 + -8x + 8x + -12x2 + 12x2 -2 + 13x + 12x2 = -8x + 8x + -12x2 + 12x2 Combine like terms: -8x + 8x = 0 -2 + 13x + 12x2 = 0 + -12x2 + 12x2 -2 + 13x + 12x2 = -12x2 + 12x2 Combine like terms: -12x2 + 12x2 = 0 -2 + 13x + 12x2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -0.1666666667 + 1.083333333x + x2 = 0 Move the constant term to the right: Add '0.1666666667' to each side of the equation. -0.1666666667 + 1.083333333x + 0.1666666667 + x2 = 0 + 0.1666666667 Reorder the terms: -0.1666666667 + 0.1666666667 + 1.083333333x + x2 = 0 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + 1.083333333x + x2 = 0 + 0.1666666667 1.083333333x + x2 = 0 + 0.1666666667 Combine like terms: 0 + 0.1666666667 = 0.1666666667 1.083333333x + x2 = 0.1666666667 The x term is 1.083333333x. Take half its coefficient (0.5416666665). Square it (0.2934027776) and add it to both sides. Add '0.2934027776' to each side of the equation. 1.083333333x + 0.2934027776 + x2 = 0.1666666667 + 0.2934027776 Reorder the terms: 0.2934027776 + 1.083333333x + x2 = 0.1666666667 + 0.2934027776 Combine like terms: 0.1666666667 + 0.2934027776 = 0.4600694443 0.2934027776 + 1.083333333x + x2 = 0.4600694443 Factor a perfect square on the left side: (x + 0.5416666665)(x + 0.5416666665) = 0.4600694443 Calculate the square root of the right side: 0.678284191 Break this problem into two subproblems by setting (x + 0.5416666665) equal to 0.678284191 and -0.678284191.Subproblem 1
x + 0.5416666665 = 0.678284191 Simplifying x + 0.5416666665 = 0.678284191 Reorder the terms: 0.5416666665 + x = 0.678284191 Solving 0.5416666665 + x = 0.678284191 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5416666665' to each side of the equation. 0.5416666665 + -0.5416666665 + x = 0.678284191 + -0.5416666665 Combine like terms: 0.5416666665 + -0.5416666665 = 0.0000000000 0.0000000000 + x = 0.678284191 + -0.5416666665 x = 0.678284191 + -0.5416666665 Combine like terms: 0.678284191 + -0.5416666665 = 0.1366175245 x = 0.1366175245 Simplifying x = 0.1366175245Subproblem 2
x + 0.5416666665 = -0.678284191 Simplifying x + 0.5416666665 = -0.678284191 Reorder the terms: 0.5416666665 + x = -0.678284191 Solving 0.5416666665 + x = -0.678284191 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5416666665' to each side of the equation. 0.5416666665 + -0.5416666665 + x = -0.678284191 + -0.5416666665 Combine like terms: 0.5416666665 + -0.5416666665 = 0.0000000000 0.0000000000 + x = -0.678284191 + -0.5416666665 x = -0.678284191 + -0.5416666665 Combine like terms: -0.678284191 + -0.5416666665 = -1.2199508575 x = -1.2199508575 Simplifying x = -1.2199508575Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.1366175245, -1.2199508575}
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