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Simplifying 67 = (2x + 3)(x + 4) Reorder the terms: 67 = (3 + 2x)(x + 4) Reorder the terms: 67 = (3 + 2x)(4 + x) Multiply (3 + 2x) * (4 + x) 67 = (3(4 + x) + 2x * (4 + x)) 67 = ((4 * 3 + x * 3) + 2x * (4 + x)) 67 = ((12 + 3x) + 2x * (4 + x)) 67 = (12 + 3x + (4 * 2x + x * 2x)) 67 = (12 + 3x + (8x + 2x2)) Combine like terms: 3x + 8x = 11x 67 = (12 + 11x + 2x2) Solving 67 = 12 + 11x + 2x2 Solving for variable 'x'. Combine like terms: 67 + -12 = 55 55 + -11x + -2x2 = 12 + 11x + 2x2 + -12 + -11x + -2x2 Reorder the terms: 55 + -11x + -2x2 = 12 + -12 + 11x + -11x + 2x2 + -2x2 Combine like terms: 12 + -12 = 0 55 + -11x + -2x2 = 0 + 11x + -11x + 2x2 + -2x2 55 + -11x + -2x2 = 11x + -11x + 2x2 + -2x2 Combine like terms: 11x + -11x = 0 55 + -11x + -2x2 = 0 + 2x2 + -2x2 55 + -11x + -2x2 = 2x2 + -2x2 Combine like terms: 2x2 + -2x2 = 0 55 + -11x + -2x2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -27.5 + 5.5x + x2 = 0 Move the constant term to the right: Add '27.5' to each side of the equation. -27.5 + 5.5x + 27.5 + x2 = 0 + 27.5 Reorder the terms: -27.5 + 27.5 + 5.5x + x2 = 0 + 27.5 Combine like terms: -27.5 + 27.5 = 0.0 0.0 + 5.5x + x2 = 0 + 27.5 5.5x + x2 = 0 + 27.5 Combine like terms: 0 + 27.5 = 27.5 5.5x + x2 = 27.5 The x term is 5.5x. Take half its coefficient (2.75). Square it (7.5625) and add it to both sides. Add '7.5625' to each side of the equation. 5.5x + 7.5625 + x2 = 27.5 + 7.5625 Reorder the terms: 7.5625 + 5.5x + x2 = 27.5 + 7.5625 Combine like terms: 27.5 + 7.5625 = 35.0625 7.5625 + 5.5x + x2 = 35.0625 Factor a perfect square on the left side: (x + 2.75)(x + 2.75) = 35.0625 Calculate the square root of the right side: 5.921359641 Break this problem into two subproblems by setting (x + 2.75) equal to 5.921359641 and -5.921359641.Subproblem 1
x + 2.75 = 5.921359641 Simplifying x + 2.75 = 5.921359641 Reorder the terms: 2.75 + x = 5.921359641 Solving 2.75 + x = 5.921359641 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.75' to each side of the equation. 2.75 + -2.75 + x = 5.921359641 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + x = 5.921359641 + -2.75 x = 5.921359641 + -2.75 Combine like terms: 5.921359641 + -2.75 = 3.171359641 x = 3.171359641 Simplifying x = 3.171359641Subproblem 2
x + 2.75 = -5.921359641 Simplifying x + 2.75 = -5.921359641 Reorder the terms: 2.75 + x = -5.921359641 Solving 2.75 + x = -5.921359641 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.75' to each side of the equation. 2.75 + -2.75 + x = -5.921359641 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + x = -5.921359641 + -2.75 x = -5.921359641 + -2.75 Combine like terms: -5.921359641 + -2.75 = -8.671359641 x = -8.671359641 Simplifying x = -8.671359641Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.171359641, -8.671359641}
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