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Simplifying 678 = 0.03x2 + -4x + 800 Reorder the terms: 678 = 800 + -4x + 0.03x2 Solving 678 = 800 + -4x + 0.03x2 Solving for variable 'x'. Combine like terms: 678 + -800 = -122 -122 + 4x + -0.03x2 = 800 + -4x + 0.03x2 + -800 + 4x + -0.03x2 Reorder the terms: -122 + 4x + -0.03x2 = 800 + -800 + -4x + 4x + 0.03x2 + -0.03x2 Combine like terms: 800 + -800 = 0 -122 + 4x + -0.03x2 = 0 + -4x + 4x + 0.03x2 + -0.03x2 -122 + 4x + -0.03x2 = -4x + 4x + 0.03x2 + -0.03x2 Combine like terms: -4x + 4x = 0 -122 + 4x + -0.03x2 = 0 + 0.03x2 + -0.03x2 -122 + 4x + -0.03x2 = 0.03x2 + -0.03x2 Combine like terms: 0.03x2 + -0.03x2 = 0.00 -122 + 4x + -0.03x2 = 0.00 Begin completing the square. Divide all terms by -0.03 the coefficient of the squared term: Divide each side by '-0.03'. 4066.666667 + -133.3333333x + x2 = 0 Move the constant term to the right: Add '-4066.666667' to each side of the equation. 4066.666667 + -133.3333333x + -4066.666667 + x2 = 0 + -4066.666667 Reorder the terms: 4066.666667 + -4066.666667 + -133.3333333x + x2 = 0 + -4066.666667 Combine like terms: 4066.666667 + -4066.666667 = 0.000000 0.000000 + -133.3333333x + x2 = 0 + -4066.666667 -133.3333333x + x2 = 0 + -4066.666667 Combine like terms: 0 + -4066.666667 = -4066.666667 -133.3333333x + x2 = -4066.666667 The x term is -133.3333333x. Take half its coefficient (-66.66666665). Square it (4444.444442) and add it to both sides. Add '4444.444442' to each side of the equation. -133.3333333x + 4444.444442 + x2 = -4066.666667 + 4444.444442 Reorder the terms: 4444.444442 + -133.3333333x + x2 = -4066.666667 + 4444.444442 Combine like terms: -4066.666667 + 4444.444442 = 377.777775 4444.444442 + -133.3333333x + x2 = 377.777775 Factor a perfect square on the left side: (x + -66.66666665)(x + -66.66666665) = 377.777775 Calculate the square root of the right side: 19.436506245 Break this problem into two subproblems by setting (x + -66.66666665) equal to 19.436506245 and -19.436506245.Subproblem 1
x + -66.66666665 = 19.436506245 Simplifying x + -66.66666665 = 19.436506245 Reorder the terms: -66.66666665 + x = 19.436506245 Solving -66.66666665 + x = 19.436506245 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '66.66666665' to each side of the equation. -66.66666665 + 66.66666665 + x = 19.436506245 + 66.66666665 Combine like terms: -66.66666665 + 66.66666665 = 0.00000000 0.00000000 + x = 19.436506245 + 66.66666665 x = 19.436506245 + 66.66666665 Combine like terms: 19.436506245 + 66.66666665 = 86.103172895 x = 86.103172895 Simplifying x = 86.103172895Subproblem 2
x + -66.66666665 = -19.436506245 Simplifying x + -66.66666665 = -19.436506245 Reorder the terms: -66.66666665 + x = -19.436506245 Solving -66.66666665 + x = -19.436506245 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '66.66666665' to each side of the equation. -66.66666665 + 66.66666665 + x = -19.436506245 + 66.66666665 Combine like terms: -66.66666665 + 66.66666665 = 0.00000000 0.00000000 + x = -19.436506245 + 66.66666665 x = -19.436506245 + 66.66666665 Combine like terms: -19.436506245 + 66.66666665 = 47.230160405 x = 47.230160405 Simplifying x = 47.230160405Solution
The solution to the problem is based on the solutions from the subproblems. x = {86.103172895, 47.230160405}
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