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Simplifying (0.5x + 1.24)(14x + 3) = 0 Reorder the terms: (1.24 + 0.5x)(14x + 3) = 0 Reorder the terms: (1.24 + 0.5x)(3 + 14x) = 0 Multiply (1.24 + 0.5x) * (3 + 14x) (1.24(3 + 14x) + 0.5x * (3 + 14x)) = 0 ((3 * 1.24 + 14x * 1.24) + 0.5x * (3 + 14x)) = 0 ((3.72 + 17.36x) + 0.5x * (3 + 14x)) = 0 (3.72 + 17.36x + (3 * 0.5x + 14x * 0.5x)) = 0 (3.72 + 17.36x + (1.5x + 7x2)) = 0 Combine like terms: 17.36x + 1.5x = 18.86x (3.72 + 18.86x + 7x2) = 0 Solving 3.72 + 18.86x + 7x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. 0.5314285714 + 2.694285714x + x2 = 0 Move the constant term to the right: Add '-0.5314285714' to each side of the equation. 0.5314285714 + 2.694285714x + -0.5314285714 + x2 = 0 + -0.5314285714 Reorder the terms: 0.5314285714 + -0.5314285714 + 2.694285714x + x2 = 0 + -0.5314285714 Combine like terms: 0.5314285714 + -0.5314285714 = 0.0000000000 0.0000000000 + 2.694285714x + x2 = 0 + -0.5314285714 2.694285714x + x2 = 0 + -0.5314285714 Combine like terms: 0 + -0.5314285714 = -0.5314285714 2.694285714x + x2 = -0.5314285714 The x term is 2.694285714x. Take half its coefficient (1.347142857). Square it (1.814793877) and add it to both sides. Add '1.814793877' to each side of the equation. 2.694285714x + 1.814793877 + x2 = -0.5314285714 + 1.814793877 Reorder the terms: 1.814793877 + 2.694285714x + x2 = -0.5314285714 + 1.814793877 Combine like terms: -0.5314285714 + 1.814793877 = 1.2833653056 1.814793877 + 2.694285714x + x2 = 1.2833653056 Factor a perfect square on the left side: (x + 1.347142857)(x + 1.347142857) = 1.2833653056 Calculate the square root of the right side: 1.132857143 Break this problem into two subproblems by setting (x + 1.347142857) equal to 1.132857143 and -1.132857143.Subproblem 1
x + 1.347142857 = 1.132857143 Simplifying x + 1.347142857 = 1.132857143 Reorder the terms: 1.347142857 + x = 1.132857143 Solving 1.347142857 + x = 1.132857143 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.347142857' to each side of the equation. 1.347142857 + -1.347142857 + x = 1.132857143 + -1.347142857 Combine like terms: 1.347142857 + -1.347142857 = 0.000000000 0.000000000 + x = 1.132857143 + -1.347142857 x = 1.132857143 + -1.347142857 Combine like terms: 1.132857143 + -1.347142857 = -0.214285714 x = -0.214285714 Simplifying x = -0.214285714Subproblem 2
x + 1.347142857 = -1.132857143 Simplifying x + 1.347142857 = -1.132857143 Reorder the terms: 1.347142857 + x = -1.132857143 Solving 1.347142857 + x = -1.132857143 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.347142857' to each side of the equation. 1.347142857 + -1.347142857 + x = -1.132857143 + -1.347142857 Combine like terms: 1.347142857 + -1.347142857 = 0.000000000 0.000000000 + x = -1.132857143 + -1.347142857 x = -1.132857143 + -1.347142857 Combine like terms: -1.132857143 + -1.347142857 = -2.48 x = -2.48 Simplifying x = -2.48Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.214285714, -2.48}
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