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Simplifying 3x2 + 253x = 120000 + -19500 + -20000 Reorder the terms: 253x + 3x2 = 120000 + -19500 + -20000 Combine like terms: 120000 + -19500 = 100500 253x + 3x2 = 100500 + -20000 Combine like terms: 100500 + -20000 = 80500 253x + 3x2 = 80500 Solving 253x + 3x2 = 80500 Solving for variable 'x'. Reorder the terms: -80500 + 253x + 3x2 = 80500 + -80500 Combine like terms: 80500 + -80500 = 0 -80500 + 253x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -26833.33333 + 84.33333333x + x2 = 0 Move the constant term to the right: Add '26833.33333' to each side of the equation. -26833.33333 + 84.33333333x + 26833.33333 + x2 = 0 + 26833.33333 Reorder the terms: -26833.33333 + 26833.33333 + 84.33333333x + x2 = 0 + 26833.33333 Combine like terms: -26833.33333 + 26833.33333 = 0.00000 0.00000 + 84.33333333x + x2 = 0 + 26833.33333 84.33333333x + x2 = 0 + 26833.33333 Combine like terms: 0 + 26833.33333 = 26833.33333 84.33333333x + x2 = 26833.33333 The x term is 84.33333333x. Take half its coefficient (42.16666667). Square it (1778.027778) and add it to both sides. Add '1778.027778' to each side of the equation. 84.33333333x + 1778.027778 + x2 = 26833.33333 + 1778.027778 Reorder the terms: 1778.027778 + 84.33333333x + x2 = 26833.33333 + 1778.027778 Combine like terms: 26833.33333 + 1778.027778 = 28611.361108 1778.027778 + 84.33333333x + x2 = 28611.361108 Factor a perfect square on the left side: (x + 42.16666667)(x + 42.16666667) = 28611.361108 Calculate the square root of the right side: 169.148931738 Break this problem into two subproblems by setting (x + 42.16666667) equal to 169.148931738 and -169.148931738.Subproblem 1
x + 42.16666667 = 169.148931738 Simplifying x + 42.16666667 = 169.148931738 Reorder the terms: 42.16666667 + x = 169.148931738 Solving 42.16666667 + x = 169.148931738 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-42.16666667' to each side of the equation. 42.16666667 + -42.16666667 + x = 169.148931738 + -42.16666667 Combine like terms: 42.16666667 + -42.16666667 = 0.00000000 0.00000000 + x = 169.148931738 + -42.16666667 x = 169.148931738 + -42.16666667 Combine like terms: 169.148931738 + -42.16666667 = 126.982265068 x = 126.982265068 Simplifying x = 126.982265068Subproblem 2
x + 42.16666667 = -169.148931738 Simplifying x + 42.16666667 = -169.148931738 Reorder the terms: 42.16666667 + x = -169.148931738 Solving 42.16666667 + x = -169.148931738 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-42.16666667' to each side of the equation. 42.16666667 + -42.16666667 + x = -169.148931738 + -42.16666667 Combine like terms: 42.16666667 + -42.16666667 = 0.00000000 0.00000000 + x = -169.148931738 + -42.16666667 x = -169.148931738 + -42.16666667 Combine like terms: -169.148931738 + -42.16666667 = -211.315598408 x = -211.315598408 Simplifying x = -211.315598408Solution
The solution to the problem is based on the solutions from the subproblems. x = {126.982265068, -211.315598408}
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