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Simplifying 0.5x(5 + -7x) = 8 + -1(4x + 6) (5 * 0.5x + -7x * 0.5x) = 8 + -1(4x + 6) (2.5x + -3.5x2) = 8 + -1(4x + 6) Reorder the terms: 2.5x + -3.5x2 = 8 + -1(6 + 4x) 2.5x + -3.5x2 = 8 + (6 * -1 + 4x * -1) 2.5x + -3.5x2 = 8 + (-6 + -4x) Combine like terms: 8 + -6 = 2 2.5x + -3.5x2 = 2 + -4x Solving 2.5x + -3.5x2 = 2 + -4x Solving for variable 'x'. Reorder the terms: -2 + 2.5x + 4x + -3.5x2 = 2 + -4x + -2 + 4x Combine like terms: 2.5x + 4x = 6.5x -2 + 6.5x + -3.5x2 = 2 + -4x + -2 + 4x Reorder the terms: -2 + 6.5x + -3.5x2 = 2 + -2 + -4x + 4x Combine like terms: 2 + -2 = 0 -2 + 6.5x + -3.5x2 = 0 + -4x + 4x -2 + 6.5x + -3.5x2 = -4x + 4x Combine like terms: -4x + 4x = 0 -2 + 6.5x + -3.5x2 = 0 Begin completing the square. Divide all terms by -3.5 the coefficient of the squared term: Divide each side by '-3.5'. 0.5714285714 + -1.857142857x + x2 = 0 Move the constant term to the right: Add '-0.5714285714' to each side of the equation. 0.5714285714 + -1.857142857x + -0.5714285714 + x2 = 0 + -0.5714285714 Reorder the terms: 0.5714285714 + -0.5714285714 + -1.857142857x + x2 = 0 + -0.5714285714 Combine like terms: 0.5714285714 + -0.5714285714 = 0.0000000000 0.0000000000 + -1.857142857x + x2 = 0 + -0.5714285714 -1.857142857x + x2 = 0 + -0.5714285714 Combine like terms: 0 + -0.5714285714 = -0.5714285714 -1.857142857x + x2 = -0.5714285714 The x term is -1.857142857x. Take half its coefficient (-0.9285714285). Square it (0.8622448978) and add it to both sides. Add '0.8622448978' to each side of the equation. -1.857142857x + 0.8622448978 + x2 = -0.5714285714 + 0.8622448978 Reorder the terms: 0.8622448978 + -1.857142857x + x2 = -0.5714285714 + 0.8622448978 Combine like terms: -0.5714285714 + 0.8622448978 = 0.2908163264 0.8622448978 + -1.857142857x + x2 = 0.2908163264 Factor a perfect square on the left side: (x + -0.9285714285)(x + -0.9285714285) = 0.2908163264 Calculate the square root of the right side: 0.539273888 Break this problem into two subproblems by setting (x + -0.9285714285) equal to 0.539273888 and -0.539273888.Subproblem 1
x + -0.9285714285 = 0.539273888 Simplifying x + -0.9285714285 = 0.539273888 Reorder the terms: -0.9285714285 + x = 0.539273888 Solving -0.9285714285 + x = 0.539273888 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.9285714285' to each side of the equation. -0.9285714285 + 0.9285714285 + x = 0.539273888 + 0.9285714285 Combine like terms: -0.9285714285 + 0.9285714285 = 0.0000000000 0.0000000000 + x = 0.539273888 + 0.9285714285 x = 0.539273888 + 0.9285714285 Combine like terms: 0.539273888 + 0.9285714285 = 1.4678453165 x = 1.4678453165 Simplifying x = 1.4678453165Subproblem 2
x + -0.9285714285 = -0.539273888 Simplifying x + -0.9285714285 = -0.539273888 Reorder the terms: -0.9285714285 + x = -0.539273888 Solving -0.9285714285 + x = -0.539273888 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.9285714285' to each side of the equation. -0.9285714285 + 0.9285714285 + x = -0.539273888 + 0.9285714285 Combine like terms: -0.9285714285 + 0.9285714285 = 0.0000000000 0.0000000000 + x = -0.539273888 + 0.9285714285 x = -0.539273888 + 0.9285714285 Combine like terms: -0.539273888 + 0.9285714285 = 0.3892975405 x = 0.3892975405 Simplifying x = 0.3892975405Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.4678453165, 0.3892975405}
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