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Simplifying 4x(7x + 2) = 23 Reorder the terms: 4x(2 + 7x) = 23 (2 * 4x + 7x * 4x) = 23 (8x + 28x2) = 23 Solving 8x + 28x2 = 23 Solving for variable 'x'. Reorder the terms: -23 + 8x + 28x2 = 23 + -23 Combine like terms: 23 + -23 = 0 -23 + 8x + 28x2 = 0 Begin completing the square. Divide all terms by 28 the coefficient of the squared term: Divide each side by '28'. -0.8214285714 + 0.2857142857x + x2 = 0 Move the constant term to the right: Add '0.8214285714' to each side of the equation. -0.8214285714 + 0.2857142857x + 0.8214285714 + x2 = 0 + 0.8214285714 Reorder the terms: -0.8214285714 + 0.8214285714 + 0.2857142857x + x2 = 0 + 0.8214285714 Combine like terms: -0.8214285714 + 0.8214285714 = 0.0000000000 0.0000000000 + 0.2857142857x + x2 = 0 + 0.8214285714 0.2857142857x + x2 = 0 + 0.8214285714 Combine like terms: 0 + 0.8214285714 = 0.8214285714 0.2857142857x + x2 = 0.8214285714 The x term is 0.2857142857x. Take half its coefficient (0.1428571429). Square it (0.02040816328) and add it to both sides. Add '0.02040816328' to each side of the equation. 0.2857142857x + 0.02040816328 + x2 = 0.8214285714 + 0.02040816328 Reorder the terms: 0.02040816328 + 0.2857142857x + x2 = 0.8214285714 + 0.02040816328 Combine like terms: 0.8214285714 + 0.02040816328 = 0.84183673468 0.02040816328 + 0.2857142857x + x2 = 0.84183673468 Factor a perfect square on the left side: (x + 0.1428571429)(x + 0.1428571429) = 0.84183673468 Calculate the square root of the right side: 0.917516613 Break this problem into two subproblems by setting (x + 0.1428571429) equal to 0.917516613 and -0.917516613.Subproblem 1
x + 0.1428571429 = 0.917516613 Simplifying x + 0.1428571429 = 0.917516613 Reorder the terms: 0.1428571429 + x = 0.917516613 Solving 0.1428571429 + x = 0.917516613 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1428571429' to each side of the equation. 0.1428571429 + -0.1428571429 + x = 0.917516613 + -0.1428571429 Combine like terms: 0.1428571429 + -0.1428571429 = 0.0000000000 0.0000000000 + x = 0.917516613 + -0.1428571429 x = 0.917516613 + -0.1428571429 Combine like terms: 0.917516613 + -0.1428571429 = 0.7746594701 x = 0.7746594701 Simplifying x = 0.7746594701Subproblem 2
x + 0.1428571429 = -0.917516613 Simplifying x + 0.1428571429 = -0.917516613 Reorder the terms: 0.1428571429 + x = -0.917516613 Solving 0.1428571429 + x = -0.917516613 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1428571429' to each side of the equation. 0.1428571429 + -0.1428571429 + x = -0.917516613 + -0.1428571429 Combine like terms: 0.1428571429 + -0.1428571429 = 0.0000000000 0.0000000000 + x = -0.917516613 + -0.1428571429 x = -0.917516613 + -0.1428571429 Combine like terms: -0.917516613 + -0.1428571429 = -1.0603737559 x = -1.0603737559 Simplifying x = -1.0603737559Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.7746594701, -1.0603737559}
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