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Simplifying 0 = 0.5x2 + 3.5x + 6 Reorder the terms: 0 = 6 + 3.5x + 0.5x2 Solving 0 = 6 + 3.5x + 0.5x2 Solving for variable 'x'. Combine like terms: 0 + -6 = -6 -6 + -3.5x + -0.5x2 = 6 + 3.5x + 0.5x2 + -6 + -3.5x + -0.5x2 Reorder the terms: -6 + -3.5x + -0.5x2 = 6 + -6 + 3.5x + -3.5x + 0.5x2 + -0.5x2 Combine like terms: 6 + -6 = 0 -6 + -3.5x + -0.5x2 = 0 + 3.5x + -3.5x + 0.5x2 + -0.5x2 -6 + -3.5x + -0.5x2 = 3.5x + -3.5x + 0.5x2 + -0.5x2 Combine like terms: 3.5x + -3.5x = 0.0 -6 + -3.5x + -0.5x2 = 0.0 + 0.5x2 + -0.5x2 -6 + -3.5x + -0.5x2 = 0.5x2 + -0.5x2 Combine like terms: 0.5x2 + -0.5x2 = 0.0 -6 + -3.5x + -0.5x2 = 0.0 Begin completing the square. Divide all terms by -0.5 the coefficient of the squared term: Divide each side by '-0.5'. 12 + 7x + x2 = 0 Move the constant term to the right: Add '-12' to each side of the equation. 12 + 7x + -12 + x2 = 0 + -12 Reorder the terms: 12 + -12 + 7x + x2 = 0 + -12 Combine like terms: 12 + -12 = 0 0 + 7x + x2 = 0 + -12 7x + x2 = 0 + -12 Combine like terms: 0 + -12 = -12 7x + x2 = -12 The x term is 7x. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7x + 12.25 + x2 = -12 + 12.25 Reorder the terms: 12.25 + 7x + x2 = -12 + 12.25 Combine like terms: -12 + 12.25 = 0.25 12.25 + 7x + x2 = 0.25 Factor a perfect square on the left side: (x + 3.5)(x + 3.5) = 0.25 Calculate the square root of the right side: 0.5 Break this problem into two subproblems by setting (x + 3.5) equal to 0.5 and -0.5.Subproblem 1
x + 3.5 = 0.5 Simplifying x + 3.5 = 0.5 Reorder the terms: 3.5 + x = 0.5 Solving 3.5 + x = 0.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + x = 0.5 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + x = 0.5 + -3.5 x = 0.5 + -3.5 Combine like terms: 0.5 + -3.5 = -3 x = -3 Simplifying x = -3Subproblem 2
x + 3.5 = -0.5 Simplifying x + 3.5 = -0.5 Reorder the terms: 3.5 + x = -0.5 Solving 3.5 + x = -0.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + x = -0.5 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + x = -0.5 + -3.5 x = -0.5 + -3.5 Combine like terms: -0.5 + -3.5 = -4 x = -4 Simplifying x = -4Solution
The solution to the problem is based on the solutions from the subproblems. x = {-3, -4}
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