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Simplifying 3x(23 + 5x) = 14 (23 * 3x + 5x * 3x) = 14 (69x + 15x2) = 14 Solving 69x + 15x2 = 14 Solving for variable 'x'. Reorder the terms: -14 + 69x + 15x2 = 14 + -14 Combine like terms: 14 + -14 = 0 -14 + 69x + 15x2 = 0 Begin completing the square. Divide all terms by 15 the coefficient of the squared term: Divide each side by '15'. -0.9333333333 + 4.6x + x2 = 0 Move the constant term to the right: Add '0.9333333333' to each side of the equation. -0.9333333333 + 4.6x + 0.9333333333 + x2 = 0 + 0.9333333333 Reorder the terms: -0.9333333333 + 0.9333333333 + 4.6x + x2 = 0 + 0.9333333333 Combine like terms: -0.9333333333 + 0.9333333333 = 0.0000000000 0.0000000000 + 4.6x + x2 = 0 + 0.9333333333 4.6x + x2 = 0 + 0.9333333333 Combine like terms: 0 + 0.9333333333 = 0.9333333333 4.6x + x2 = 0.9333333333 The x term is 4.6x. Take half its coefficient (2.3). Square it (5.29) and add it to both sides. Add '5.29' to each side of the equation. 4.6x + 5.29 + x2 = 0.9333333333 + 5.29 Reorder the terms: 5.29 + 4.6x + x2 = 0.9333333333 + 5.29 Combine like terms: 0.9333333333 + 5.29 = 6.2233333333 5.29 + 4.6x + x2 = 6.2233333333 Factor a perfect square on the left side: (x + 2.3)(x + 2.3) = 6.2233333333 Calculate the square root of the right side: 2.494660966 Break this problem into two subproblems by setting (x + 2.3) equal to 2.494660966 and -2.494660966.Subproblem 1
x + 2.3 = 2.494660966 Simplifying x + 2.3 = 2.494660966 Reorder the terms: 2.3 + x = 2.494660966 Solving 2.3 + x = 2.494660966 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.3' to each side of the equation. 2.3 + -2.3 + x = 2.494660966 + -2.3 Combine like terms: 2.3 + -2.3 = 0.0 0.0 + x = 2.494660966 + -2.3 x = 2.494660966 + -2.3 Combine like terms: 2.494660966 + -2.3 = 0.194660966 x = 0.194660966 Simplifying x = 0.194660966Subproblem 2
x + 2.3 = -2.494660966 Simplifying x + 2.3 = -2.494660966 Reorder the terms: 2.3 + x = -2.494660966 Solving 2.3 + x = -2.494660966 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.3' to each side of the equation. 2.3 + -2.3 + x = -2.494660966 + -2.3 Combine like terms: 2.3 + -2.3 = 0.0 0.0 + x = -2.494660966 + -2.3 x = -2.494660966 + -2.3 Combine like terms: -2.494660966 + -2.3 = -4.794660966 x = -4.794660966 Simplifying x = -4.794660966Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.194660966, -4.794660966}
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