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Simplifying 0 = 0.5x2 + 5x + 12 Reorder the terms: 0 = 12 + 5x + 0.5x2 Solving 0 = 12 + 5x + 0.5x2 Solving for variable 'x'. Combine like terms: 0 + -12 = -12 -12 + -5x + -0.5x2 = 12 + 5x + 0.5x2 + -12 + -5x + -0.5x2 Reorder the terms: -12 + -5x + -0.5x2 = 12 + -12 + 5x + -5x + 0.5x2 + -0.5x2 Combine like terms: 12 + -12 = 0 -12 + -5x + -0.5x2 = 0 + 5x + -5x + 0.5x2 + -0.5x2 -12 + -5x + -0.5x2 = 5x + -5x + 0.5x2 + -0.5x2 Combine like terms: 5x + -5x = 0 -12 + -5x + -0.5x2 = 0 + 0.5x2 + -0.5x2 -12 + -5x + -0.5x2 = 0.5x2 + -0.5x2 Combine like terms: 0.5x2 + -0.5x2 = 0.0 -12 + -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'. 24 + 10x + x2 = 0 Move the constant term to the right: Add '-24' to each side of the equation. 24 + 10x + -24 + x2 = 0 + -24 Reorder the terms: 24 + -24 + 10x + x2 = 0 + -24 Combine like terms: 24 + -24 = 0 0 + 10x + x2 = 0 + -24 10x + x2 = 0 + -24 Combine like terms: 0 + -24 = -24 10x + x2 = -24 The x term is 10x. Take half its coefficient (5). Square it (25) and add it to both sides. Add '25' to each side of the equation. 10x + 25 + x2 = -24 + 25 Reorder the terms: 25 + 10x + x2 = -24 + 25 Combine like terms: -24 + 25 = 1 25 + 10x + x2 = 1 Factor a perfect square on the left side: (x + 5)(x + 5) = 1 Calculate the square root of the right side: 1 Break this problem into two subproblems by setting (x + 5) equal to 1 and -1.Subproblem 1
x + 5 = 1 Simplifying x + 5 = 1 Reorder the terms: 5 + x = 1 Solving 5 + x = 1 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + x = 1 + -5 Combine like terms: 5 + -5 = 0 0 + x = 1 + -5 x = 1 + -5 Combine like terms: 1 + -5 = -4 x = -4 Simplifying x = -4Subproblem 2
x + 5 = -1 Simplifying x + 5 = -1 Reorder the terms: 5 + x = -1 Solving 5 + x = -1 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + x = -1 + -5 Combine like terms: 5 + -5 = 0 0 + x = -1 + -5 x = -1 + -5 Combine like terms: -1 + -5 = -6 x = -6 Simplifying x = -6Solution
The solution to the problem is based on the solutions from the subproblems. x = {-4, -6}
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