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Simplifying 3x(4x + 5) + -1(2x + 9) = 0 Reorder the terms: 3x(5 + 4x) + -1(2x + 9) = 0 (5 * 3x + 4x * 3x) + -1(2x + 9) = 0 (15x + 12x2) + -1(2x + 9) = 0 Reorder the terms: 15x + 12x2 + -1(9 + 2x) = 0 15x + 12x2 + (9 * -1 + 2x * -1) = 0 15x + 12x2 + (-9 + -2x) = 0 Reorder the terms: -9 + 15x + -2x + 12x2 = 0 Combine like terms: 15x + -2x = 13x -9 + 13x + 12x2 = 0 Solving -9 + 13x + 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'. -0.75 + 1.083333333x + x2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + 1.083333333x + 0.75 + x2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + 1.083333333x + x2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + 1.083333333x + x2 = 0 + 0.75 1.083333333x + x2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 1.083333333x + x2 = 0.75 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 = 0.75 + 0.2934027776 Reorder the terms: 0.2934027776 + 1.083333333x + x2 = 0.75 + 0.2934027776 Combine like terms: 0.75 + 0.2934027776 = 1.0434027776 0.2934027776 + 1.083333333x + x2 = 1.0434027776 Factor a perfect square on the left side: (x + 0.5416666665)(x + 0.5416666665) = 1.0434027776 Calculate the square root of the right side: 1.021470889 Break this problem into two subproblems by setting (x + 0.5416666665) equal to 1.021470889 and -1.021470889.Subproblem 1
x + 0.5416666665 = 1.021470889 Simplifying x + 0.5416666665 = 1.021470889 Reorder the terms: 0.5416666665 + x = 1.021470889 Solving 0.5416666665 + x = 1.021470889 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 = 1.021470889 + -0.5416666665 Combine like terms: 0.5416666665 + -0.5416666665 = 0.0000000000 0.0000000000 + x = 1.021470889 + -0.5416666665 x = 1.021470889 + -0.5416666665 Combine like terms: 1.021470889 + -0.5416666665 = 0.4798042225 x = 0.4798042225 Simplifying x = 0.4798042225Subproblem 2
x + 0.5416666665 = -1.021470889 Simplifying x + 0.5416666665 = -1.021470889 Reorder the terms: 0.5416666665 + x = -1.021470889 Solving 0.5416666665 + x = -1.021470889 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 = -1.021470889 + -0.5416666665 Combine like terms: 0.5416666665 + -0.5416666665 = 0.0000000000 0.0000000000 + x = -1.021470889 + -0.5416666665 x = -1.021470889 + -0.5416666665 Combine like terms: -1.021470889 + -0.5416666665 = -1.5631375555 x = -1.5631375555 Simplifying x = -1.5631375555Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.4798042225, -1.5631375555}
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