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Simplifying (5x + 3)(2x + -1) + -3(x + 4) = 0 Reorder the terms: (3 + 5x)(2x + -1) + -3(x + 4) = 0 Reorder the terms: (3 + 5x)(-1 + 2x) + -3(x + 4) = 0 Multiply (3 + 5x) * (-1 + 2x) (3(-1 + 2x) + 5x * (-1 + 2x)) + -3(x + 4) = 0 ((-1 * 3 + 2x * 3) + 5x * (-1 + 2x)) + -3(x + 4) = 0 ((-3 + 6x) + 5x * (-1 + 2x)) + -3(x + 4) = 0 (-3 + 6x + (-1 * 5x + 2x * 5x)) + -3(x + 4) = 0 (-3 + 6x + (-5x + 10x2)) + -3(x + 4) = 0 Combine like terms: 6x + -5x = 1x (-3 + 1x + 10x2) + -3(x + 4) = 0 Reorder the terms: -3 + 1x + 10x2 + -3(4 + x) = 0 -3 + 1x + 10x2 + (4 * -3 + x * -3) = 0 -3 + 1x + 10x2 + (-12 + -3x) = 0 Reorder the terms: -3 + -12 + 1x + -3x + 10x2 = 0 Combine like terms: -3 + -12 = -15 -15 + 1x + -3x + 10x2 = 0 Combine like terms: 1x + -3x = -2x -15 + -2x + 10x2 = 0 Solving -15 + -2x + 10x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. -1.5 + -0.2x + x2 = 0.0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + -0.2x + 1.5 + x2 = 0.0 + 1.5 Reorder the terms: -1.5 + 1.5 + -0.2x + x2 = 0.0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + -0.2x + x2 = 0.0 + 1.5 -0.2x + x2 = 0.0 + 1.5 Combine like terms: 0.0 + 1.5 = 1.5 -0.2x + x2 = 1.5 The x term is -0.2x. Take half its coefficient (-0.1). Square it (0.01) and add it to both sides. Add '0.01' to each side of the equation. -0.2x + 0.01 + x2 = 1.5 + 0.01 Reorder the terms: 0.01 + -0.2x + x2 = 1.5 + 0.01 Combine like terms: 1.5 + 0.01 = 1.51 0.01 + -0.2x + x2 = 1.51 Factor a perfect square on the left side: (x + -0.1)(x + -0.1) = 1.51 Calculate the square root of the right side: 1.228820573 Break this problem into two subproblems by setting (x + -0.1) equal to 1.228820573 and -1.228820573.Subproblem 1
x + -0.1 = 1.228820573 Simplifying x + -0.1 = 1.228820573 Reorder the terms: -0.1 + x = 1.228820573 Solving -0.1 + x = 1.228820573 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1' to each side of the equation. -0.1 + 0.1 + x = 1.228820573 + 0.1 Combine like terms: -0.1 + 0.1 = 0.0 0.0 + x = 1.228820573 + 0.1 x = 1.228820573 + 0.1 Combine like terms: 1.228820573 + 0.1 = 1.328820573 x = 1.328820573 Simplifying x = 1.328820573Subproblem 2
x + -0.1 = -1.228820573 Simplifying x + -0.1 = -1.228820573 Reorder the terms: -0.1 + x = -1.228820573 Solving -0.1 + x = -1.228820573 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1' to each side of the equation. -0.1 + 0.1 + x = -1.228820573 + 0.1 Combine like terms: -0.1 + 0.1 = 0.0 0.0 + x = -1.228820573 + 0.1 x = -1.228820573 + 0.1 Combine like terms: -1.228820573 + 0.1 = -1.128820573 x = -1.128820573 Simplifying x = -1.128820573Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.328820573, -1.128820573}
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