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Simplifying 3x2 + 10x + 3 = 19 + -4x2 + 16 Reorder the terms: 3 + 10x + 3x2 = 19 + -4x2 + 16 Reorder the terms: 3 + 10x + 3x2 = 19 + 16 + -4x2 Combine like terms: 19 + 16 = 35 3 + 10x + 3x2 = 35 + -4x2 Solving 3 + 10x + 3x2 = 35 + -4x2 Solving for variable 'x'. Reorder the terms: 3 + -35 + 10x + 3x2 + 4x2 = 35 + -4x2 + -35 + 4x2 Combine like terms: 3 + -35 = -32 -32 + 10x + 3x2 + 4x2 = 35 + -4x2 + -35 + 4x2 Combine like terms: 3x2 + 4x2 = 7x2 -32 + 10x + 7x2 = 35 + -4x2 + -35 + 4x2 Reorder the terms: -32 + 10x + 7x2 = 35 + -35 + -4x2 + 4x2 Combine like terms: 35 + -35 = 0 -32 + 10x + 7x2 = 0 + -4x2 + 4x2 -32 + 10x + 7x2 = -4x2 + 4x2 Combine like terms: -4x2 + 4x2 = 0 -32 + 10x + 7x2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -4.571428571 + 1.428571429x + x2 = 0 Move the constant term to the right: Add '4.571428571' to each side of the equation. -4.571428571 + 1.428571429x + 4.571428571 + x2 = 0 + 4.571428571 Reorder the terms: -4.571428571 + 4.571428571 + 1.428571429x + x2 = 0 + 4.571428571 Combine like terms: -4.571428571 + 4.571428571 = 0.000000000 0.000000000 + 1.428571429x + x2 = 0 + 4.571428571 1.428571429x + x2 = 0 + 4.571428571 Combine like terms: 0 + 4.571428571 = 4.571428571 1.428571429x + x2 = 4.571428571 The x term is 1.428571429x. Take half its coefficient (0.7142857145). Square it (0.5102040819) and add it to both sides. Add '0.5102040819' to each side of the equation. 1.428571429x + 0.5102040819 + x2 = 4.571428571 + 0.5102040819 Reorder the terms: 0.5102040819 + 1.428571429x + x2 = 4.571428571 + 0.5102040819 Combine like terms: 4.571428571 + 0.5102040819 = 5.0816326529 0.5102040819 + 1.428571429x + x2 = 5.0816326529 Factor a perfect square on the left side: (x + 0.7142857145)(x + 0.7142857145) = 5.0816326529 Calculate the square root of the right side: 2.254247691 Break this problem into two subproblems by setting (x + 0.7142857145) equal to 2.254247691 and -2.254247691.Subproblem 1
x + 0.7142857145 = 2.254247691 Simplifying x + 0.7142857145 = 2.254247691 Reorder the terms: 0.7142857145 + x = 2.254247691 Solving 0.7142857145 + x = 2.254247691 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.7142857145' to each side of the equation. 0.7142857145 + -0.7142857145 + x = 2.254247691 + -0.7142857145 Combine like terms: 0.7142857145 + -0.7142857145 = 0.0000000000 0.0000000000 + x = 2.254247691 + -0.7142857145 x = 2.254247691 + -0.7142857145 Combine like terms: 2.254247691 + -0.7142857145 = 1.5399619765 x = 1.5399619765 Simplifying x = 1.5399619765Subproblem 2
x + 0.7142857145 = -2.254247691 Simplifying x + 0.7142857145 = -2.254247691 Reorder the terms: 0.7142857145 + x = -2.254247691 Solving 0.7142857145 + x = -2.254247691 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.7142857145' to each side of the equation. 0.7142857145 + -0.7142857145 + x = -2.254247691 + -0.7142857145 Combine like terms: 0.7142857145 + -0.7142857145 = 0.0000000000 0.0000000000 + x = -2.254247691 + -0.7142857145 x = -2.254247691 + -0.7142857145 Combine like terms: -2.254247691 + -0.7142857145 = -2.9685334055 x = -2.9685334055 Simplifying x = -2.9685334055Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.5399619765, -2.9685334055}
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