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Simplifying (2x + -3)(x + 2) = 7 Reorder the terms: (-3 + 2x)(x + 2) = 7 Reorder the terms: (-3 + 2x)(2 + x) = 7 Multiply (-3 + 2x) * (2 + x) (-3(2 + x) + 2x * (2 + x)) = 7 ((2 * -3 + x * -3) + 2x * (2 + x)) = 7 ((-6 + -3x) + 2x * (2 + x)) = 7 (-6 + -3x + (2 * 2x + x * 2x)) = 7 (-6 + -3x + (4x + 2x2)) = 7 Combine like terms: -3x + 4x = 1x (-6 + 1x + 2x2) = 7 Solving -6 + 1x + 2x2 = 7 Solving for variable 'x'. Reorder the terms: -6 + -7 + 1x + 2x2 = 7 + -7 Combine like terms: -6 + -7 = -13 -13 + 1x + 2x2 = 7 + -7 Combine like terms: 7 + -7 = 0 -13 + 1x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -6.5 + 0.5x + x2 = 0 Move the constant term to the right: Add '6.5' to each side of the equation. -6.5 + 0.5x + 6.5 + x2 = 0 + 6.5 Reorder the terms: -6.5 + 6.5 + 0.5x + x2 = 0 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + 0.5x + x2 = 0 + 6.5 0.5x + x2 = 0 + 6.5 Combine like terms: 0 + 6.5 = 6.5 0.5x + x2 = 6.5 The x term is 0.5x. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5x + 0.0625 + x2 = 6.5 + 0.0625 Reorder the terms: 0.0625 + 0.5x + x2 = 6.5 + 0.0625 Combine like terms: 6.5 + 0.0625 = 6.5625 0.0625 + 0.5x + x2 = 6.5625 Factor a perfect square on the left side: (x + 0.25)(x + 0.25) = 6.5625 Calculate the square root of the right side: 2.561737691 Break this problem into two subproblems by setting (x + 0.25) equal to 2.561737691 and -2.561737691.Subproblem 1
x + 0.25 = 2.561737691 Simplifying x + 0.25 = 2.561737691 Reorder the terms: 0.25 + x = 2.561737691 Solving 0.25 + x = 2.561737691 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = 2.561737691 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 2.561737691 + -0.25 x = 2.561737691 + -0.25 Combine like terms: 2.561737691 + -0.25 = 2.311737691 x = 2.311737691 Simplifying x = 2.311737691Subproblem 2
x + 0.25 = -2.561737691 Simplifying x + 0.25 = -2.561737691 Reorder the terms: 0.25 + x = -2.561737691 Solving 0.25 + x = -2.561737691 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = -2.561737691 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -2.561737691 + -0.25 x = -2.561737691 + -0.25 Combine like terms: -2.561737691 + -0.25 = -2.811737691 x = -2.811737691 Simplifying x = -2.811737691Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.311737691, -2.811737691}
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