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Simplifying 78 = 0.5x2 + -12x + 150 Reorder the terms: 78 = 150 + -12x + 0.5x2 Solving 78 = 150 + -12x + 0.5x2 Solving for variable 'x'. Combine like terms: 78 + -150 = -72 -72 + 12x + -0.5x2 = 150 + -12x + 0.5x2 + -150 + 12x + -0.5x2 Reorder the terms: -72 + 12x + -0.5x2 = 150 + -150 + -12x + 12x + 0.5x2 + -0.5x2 Combine like terms: 150 + -150 = 0 -72 + 12x + -0.5x2 = 0 + -12x + 12x + 0.5x2 + -0.5x2 -72 + 12x + -0.5x2 = -12x + 12x + 0.5x2 + -0.5x2 Combine like terms: -12x + 12x = 0 -72 + 12x + -0.5x2 = 0 + 0.5x2 + -0.5x2 -72 + 12x + -0.5x2 = 0.5x2 + -0.5x2 Combine like terms: 0.5x2 + -0.5x2 = 0.0 -72 + 12x + -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'. 144 + -24x + x2 = 0 Move the constant term to the right: Add '-144' to each side of the equation. 144 + -24x + -144 + x2 = 0 + -144 Reorder the terms: 144 + -144 + -24x + x2 = 0 + -144 Combine like terms: 144 + -144 = 0 0 + -24x + x2 = 0 + -144 -24x + x2 = 0 + -144 Combine like terms: 0 + -144 = -144 -24x + x2 = -144 The x term is -24x. Take half its coefficient (-12). Square it (144) and add it to both sides. Add '144' to each side of the equation. -24x + 144 + x2 = -144 + 144 Reorder the terms: 144 + -24x + x2 = -144 + 144 Combine like terms: -144 + 144 = 0 144 + -24x + x2 = 0 Factor a perfect square on the left side: (x + -12)(x + -12) = 0 Calculate the square root of the right side: 0 Break this problem into two subproblems by setting (x + -12) equal to 0 and 0.Subproblem 1
x + -12 = 0 Simplifying x + -12 = 0 Reorder the terms: -12 + x = 0 Solving -12 + x = 0 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '12' to each side of the equation. -12 + 12 + x = 0 + 12 Combine like terms: -12 + 12 = 0 0 + x = 0 + 12 x = 0 + 12 Combine like terms: 0 + 12 = 12 x = 12 Simplifying x = 12Subproblem 2
x + -12 = 0 Simplifying x + -12 = 0 Reorder the terms: -12 + x = 0 Solving -12 + x = 0 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '12' to each side of the equation. -12 + 12 + x = 0 + 12 Combine like terms: -12 + 12 = 0 0 + x = 0 + 12 x = 0 + 12 Combine like terms: 0 + 12 = 12 x = 12 Simplifying x = 12Solution
The solution to the problem is based on the solutions from the subproblems. x = {12, 12}
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