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Simplifying 4x(3x + 5) = 40 + 7x Reorder the terms: 4x(5 + 3x) = 40 + 7x (5 * 4x + 3x * 4x) = 40 + 7x (20x + 12x2) = 40 + 7x Solving 20x + 12x2 = 40 + 7x Solving for variable 'x'. Reorder the terms: -40 + 20x + -7x + 12x2 = 40 + 7x + -40 + -7x Combine like terms: 20x + -7x = 13x -40 + 13x + 12x2 = 40 + 7x + -40 + -7x Reorder the terms: -40 + 13x + 12x2 = 40 + -40 + 7x + -7x Combine like terms: 40 + -40 = 0 -40 + 13x + 12x2 = 0 + 7x + -7x -40 + 13x + 12x2 = 7x + -7x Combine like terms: 7x + -7x = 0 -40 + 13x + 12x2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -3.333333333 + 1.083333333x + x2 = 0 Move the constant term to the right: Add '3.333333333' to each side of the equation. -3.333333333 + 1.083333333x + 3.333333333 + x2 = 0 + 3.333333333 Reorder the terms: -3.333333333 + 3.333333333 + 1.083333333x + x2 = 0 + 3.333333333 Combine like terms: -3.333333333 + 3.333333333 = 0.000000000 0.000000000 + 1.083333333x + x2 = 0 + 3.333333333 1.083333333x + x2 = 0 + 3.333333333 Combine like terms: 0 + 3.333333333 = 3.333333333 1.083333333x + x2 = 3.333333333 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 = 3.333333333 + 0.2934027776 Reorder the terms: 0.2934027776 + 1.083333333x + x2 = 3.333333333 + 0.2934027776 Combine like terms: 3.333333333 + 0.2934027776 = 3.6267361106 0.2934027776 + 1.083333333x + x2 = 3.6267361106 Factor a perfect square on the left side: (x + 0.5416666665)(x + 0.5416666665) = 3.6267361106 Calculate the square root of the right side: 1.904399147 Break this problem into two subproblems by setting (x + 0.5416666665) equal to 1.904399147 and -1.904399147.Subproblem 1
x + 0.5416666665 = 1.904399147 Simplifying x + 0.5416666665 = 1.904399147 Reorder the terms: 0.5416666665 + x = 1.904399147 Solving 0.5416666665 + x = 1.904399147 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 = 1.904399147 + -0.5416666665 Combine like terms: 0.5416666665 + -0.5416666665 = 0.0000000000 0.0000000000 + x = 1.904399147 + -0.5416666665 x = 1.904399147 + -0.5416666665 Combine like terms: 1.904399147 + -0.5416666665 = 1.3627324805 x = 1.3627324805 Simplifying x = 1.3627324805Subproblem 2
x + 0.5416666665 = -1.904399147 Simplifying x + 0.5416666665 = -1.904399147 Reorder the terms: 0.5416666665 + x = -1.904399147 Solving 0.5416666665 + x = -1.904399147 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 = -1.904399147 + -0.5416666665 Combine like terms: 0.5416666665 + -0.5416666665 = 0.0000000000 0.0000000000 + x = -1.904399147 + -0.5416666665 x = -1.904399147 + -0.5416666665 Combine like terms: -1.904399147 + -0.5416666665 = -2.4460658135 x = -2.4460658135 Simplifying x = -2.4460658135Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.3627324805, -2.4460658135}
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