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Simplifying 4x2 + -6x + -2x2 + 19x = 180 Reorder the terms: -6x + 19x + 4x2 + -2x2 = 180 Combine like terms: -6x + 19x = 13x 13x + 4x2 + -2x2 = 180 Combine like terms: 4x2 + -2x2 = 2x2 13x + 2x2 = 180 Solving 13x + 2x2 = 180 Solving for variable 'x'. Reorder the terms: -180 + 13x + 2x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -180 + 13x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -90 + 6.5x + x2 = 0 Move the constant term to the right: Add '90' to each side of the equation. -90 + 6.5x + 90 + x2 = 0 + 90 Reorder the terms: -90 + 90 + 6.5x + x2 = 0 + 90 Combine like terms: -90 + 90 = 0 0 + 6.5x + x2 = 0 + 90 6.5x + x2 = 0 + 90 Combine like terms: 0 + 90 = 90 6.5x + x2 = 90 The x term is 6.5x. Take half its coefficient (3.25). Square it (10.5625) and add it to both sides. Add '10.5625' to each side of the equation. 6.5x + 10.5625 + x2 = 90 + 10.5625 Reorder the terms: 10.5625 + 6.5x + x2 = 90 + 10.5625 Combine like terms: 90 + 10.5625 = 100.5625 10.5625 + 6.5x + x2 = 100.5625 Factor a perfect square on the left side: (x + 3.25)(x + 3.25) = 100.5625 Calculate the square root of the right side: 10.02808556 Break this problem into two subproblems by setting (x + 3.25) equal to 10.02808556 and -10.02808556.Subproblem 1
x + 3.25 = 10.02808556 Simplifying x + 3.25 = 10.02808556 Reorder the terms: 3.25 + x = 10.02808556 Solving 3.25 + x = 10.02808556 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.25' to each side of the equation. 3.25 + -3.25 + x = 10.02808556 + -3.25 Combine like terms: 3.25 + -3.25 = 0.00 0.00 + x = 10.02808556 + -3.25 x = 10.02808556 + -3.25 Combine like terms: 10.02808556 + -3.25 = 6.77808556 x = 6.77808556 Simplifying x = 6.77808556Subproblem 2
x + 3.25 = -10.02808556 Simplifying x + 3.25 = -10.02808556 Reorder the terms: 3.25 + x = -10.02808556 Solving 3.25 + x = -10.02808556 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.25' to each side of the equation. 3.25 + -3.25 + x = -10.02808556 + -3.25 Combine like terms: 3.25 + -3.25 = 0.00 0.00 + x = -10.02808556 + -3.25 x = -10.02808556 + -3.25 Combine like terms: -10.02808556 + -3.25 = -13.27808556 x = -13.27808556 Simplifying x = -13.27808556Solution
The solution to the problem is based on the solutions from the subproblems. x = {6.77808556, -13.27808556}
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