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Simplifying 7x2 + 21x + 6 = 0 Reorder the terms: 6 + 21x + 7x2 = 0 Solving 6 + 21x + 7x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. 0.8571428571 + 3x + x2 = 0 Move the constant term to the right: Add '-0.8571428571' to each side of the equation. 0.8571428571 + 3x + -0.8571428571 + x2 = 0 + -0.8571428571 Reorder the terms: 0.8571428571 + -0.8571428571 + 3x + x2 = 0 + -0.8571428571 Combine like terms: 0.8571428571 + -0.8571428571 = 0.0000000000 0.0000000000 + 3x + x2 = 0 + -0.8571428571 3x + x2 = 0 + -0.8571428571 Combine like terms: 0 + -0.8571428571 = -0.8571428571 3x + x2 = -0.8571428571 The x term is 3x. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3x + 2.25 + x2 = -0.8571428571 + 2.25 Reorder the terms: 2.25 + 3x + x2 = -0.8571428571 + 2.25 Combine like terms: -0.8571428571 + 2.25 = 1.3928571429 2.25 + 3x + x2 = 1.3928571429 Factor a perfect square on the left side: (x + 1.5)(x + 1.5) = 1.3928571429 Calculate the square root of the right side: 1.180193689 Break this problem into two subproblems by setting (x + 1.5) equal to 1.180193689 and -1.180193689.Subproblem 1
x + 1.5 = 1.180193689 Simplifying x + 1.5 = 1.180193689 Reorder the terms: 1.5 + x = 1.180193689 Solving 1.5 + x = 1.180193689 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = 1.180193689 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 1.180193689 + -1.5 x = 1.180193689 + -1.5 Combine like terms: 1.180193689 + -1.5 = -0.319806311 x = -0.319806311 Simplifying x = -0.319806311Subproblem 2
x + 1.5 = -1.180193689 Simplifying x + 1.5 = -1.180193689 Reorder the terms: 1.5 + x = -1.180193689 Solving 1.5 + x = -1.180193689 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = -1.180193689 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -1.180193689 + -1.5 x = -1.180193689 + -1.5 Combine like terms: -1.180193689 + -1.5 = -2.680193689 x = -2.680193689 Simplifying x = -2.680193689Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.319806311, -2.680193689}
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