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Simplifying (7x + -14)(4x + 19) = 180 Reorder the terms: (-14 + 7x)(4x + 19) = 180 Reorder the terms: (-14 + 7x)(19 + 4x) = 180 Multiply (-14 + 7x) * (19 + 4x) (-14(19 + 4x) + 7x * (19 + 4x)) = 180 ((19 * -14 + 4x * -14) + 7x * (19 + 4x)) = 180 ((-266 + -56x) + 7x * (19 + 4x)) = 180 (-266 + -56x + (19 * 7x + 4x * 7x)) = 180 (-266 + -56x + (133x + 28x2)) = 180 Combine like terms: -56x + 133x = 77x (-266 + 77x + 28x2) = 180 Solving -266 + 77x + 28x2 = 180 Solving for variable 'x'. Reorder the terms: -266 + -180 + 77x + 28x2 = 180 + -180 Combine like terms: -266 + -180 = -446 -446 + 77x + 28x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -446 + 77x + 28x2 = 0 Begin completing the square. Divide all terms by 28 the coefficient of the squared term: Divide each side by '28'. -15.92857143 + 2.75x + x2 = 0 Move the constant term to the right: Add '15.92857143' to each side of the equation. -15.92857143 + 2.75x + 15.92857143 + x2 = 0 + 15.92857143 Reorder the terms: -15.92857143 + 15.92857143 + 2.75x + x2 = 0 + 15.92857143 Combine like terms: -15.92857143 + 15.92857143 = 0.00000000 0.00000000 + 2.75x + x2 = 0 + 15.92857143 2.75x + x2 = 0 + 15.92857143 Combine like terms: 0 + 15.92857143 = 15.92857143 2.75x + x2 = 15.92857143 The x term is 2.75x. Take half its coefficient (1.375). Square it (1.890625) and add it to both sides. Add '1.890625' to each side of the equation. 2.75x + 1.890625 + x2 = 15.92857143 + 1.890625 Reorder the terms: 1.890625 + 2.75x + x2 = 15.92857143 + 1.890625 Combine like terms: 15.92857143 + 1.890625 = 17.81919643 1.890625 + 2.75x + x2 = 17.81919643 Factor a perfect square on the left side: (x + 1.375)(x + 1.375) = 17.81919643 Calculate the square root of the right side: 4.221279004 Break this problem into two subproblems by setting (x + 1.375) equal to 4.221279004 and -4.221279004.Subproblem 1
x + 1.375 = 4.221279004 Simplifying x + 1.375 = 4.221279004 Reorder the terms: 1.375 + x = 4.221279004 Solving 1.375 + x = 4.221279004 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.375' to each side of the equation. 1.375 + -1.375 + x = 4.221279004 + -1.375 Combine like terms: 1.375 + -1.375 = 0.000 0.000 + x = 4.221279004 + -1.375 x = 4.221279004 + -1.375 Combine like terms: 4.221279004 + -1.375 = 2.846279004 x = 2.846279004 Simplifying x = 2.846279004Subproblem 2
x + 1.375 = -4.221279004 Simplifying x + 1.375 = -4.221279004 Reorder the terms: 1.375 + x = -4.221279004 Solving 1.375 + x = -4.221279004 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.375' to each side of the equation. 1.375 + -1.375 + x = -4.221279004 + -1.375 Combine like terms: 1.375 + -1.375 = 0.000 0.000 + x = -4.221279004 + -1.375 x = -4.221279004 + -1.375 Combine like terms: -4.221279004 + -1.375 = -5.596279004 x = -5.596279004 Simplifying x = -5.596279004Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.846279004, -5.596279004}
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