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Simplifying (3x + 11)(4x + -5) = 180 Reorder the terms: (11 + 3x)(4x + -5) = 180 Reorder the terms: (11 + 3x)(-5 + 4x) = 180 Multiply (11 + 3x) * (-5 + 4x) (11(-5 + 4x) + 3x * (-5 + 4x)) = 180 ((-5 * 11 + 4x * 11) + 3x * (-5 + 4x)) = 180 ((-55 + 44x) + 3x * (-5 + 4x)) = 180 (-55 + 44x + (-5 * 3x + 4x * 3x)) = 180 (-55 + 44x + (-15x + 12x2)) = 180 Combine like terms: 44x + -15x = 29x (-55 + 29x + 12x2) = 180 Solving -55 + 29x + 12x2 = 180 Solving for variable 'x'. Reorder the terms: -55 + -180 + 29x + 12x2 = 180 + -180 Combine like terms: -55 + -180 = -235 -235 + 29x + 12x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -235 + 29x + 12x2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -19.58333333 + 2.416666667x + x2 = 0 Move the constant term to the right: Add '19.58333333' to each side of the equation. -19.58333333 + 2.416666667x + 19.58333333 + x2 = 0 + 19.58333333 Reorder the terms: -19.58333333 + 19.58333333 + 2.416666667x + x2 = 0 + 19.58333333 Combine like terms: -19.58333333 + 19.58333333 = 0.00000000 0.00000000 + 2.416666667x + x2 = 0 + 19.58333333 2.416666667x + x2 = 0 + 19.58333333 Combine like terms: 0 + 19.58333333 = 19.58333333 2.416666667x + x2 = 19.58333333 The x term is 2.416666667x. Take half its coefficient (1.208333334). Square it (1.460069446) and add it to both sides. Add '1.460069446' to each side of the equation. 2.416666667x + 1.460069446 + x2 = 19.58333333 + 1.460069446 Reorder the terms: 1.460069446 + 2.416666667x + x2 = 19.58333333 + 1.460069446 Combine like terms: 19.58333333 + 1.460069446 = 21.043402776 1.460069446 + 2.416666667x + x2 = 21.043402776 Factor a perfect square on the left side: (x + 1.208333334)(x + 1.208333334) = 21.043402776 Calculate the square root of the right side: 4.587308882 Break this problem into two subproblems by setting (x + 1.208333334) equal to 4.587308882 and -4.587308882.Subproblem 1
x + 1.208333334 = 4.587308882 Simplifying x + 1.208333334 = 4.587308882 Reorder the terms: 1.208333334 + x = 4.587308882 Solving 1.208333334 + x = 4.587308882 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.208333334' to each side of the equation. 1.208333334 + -1.208333334 + x = 4.587308882 + -1.208333334 Combine like terms: 1.208333334 + -1.208333334 = 0.000000000 0.000000000 + x = 4.587308882 + -1.208333334 x = 4.587308882 + -1.208333334 Combine like terms: 4.587308882 + -1.208333334 = 3.378975548 x = 3.378975548 Simplifying x = 3.378975548Subproblem 2
x + 1.208333334 = -4.587308882 Simplifying x + 1.208333334 = -4.587308882 Reorder the terms: 1.208333334 + x = -4.587308882 Solving 1.208333334 + x = -4.587308882 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.208333334' to each side of the equation. 1.208333334 + -1.208333334 + x = -4.587308882 + -1.208333334 Combine like terms: 1.208333334 + -1.208333334 = 0.000000000 0.000000000 + x = -4.587308882 + -1.208333334 x = -4.587308882 + -1.208333334 Combine like terms: -4.587308882 + -1.208333334 = -5.795642216 x = -5.795642216 Simplifying x = -5.795642216Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.378975548, -5.795642216}
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