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Simplifying 1000 + -0.4x2 = 42x Solving 1000 + -0.4x2 = 42x Solving for variable 'x'. Reorder the terms: 1000 + -42x + -0.4x2 = 42x + -42x Combine like terms: 42x + -42x = 0 1000 + -42x + -0.4x2 = 0 Begin completing the square. Divide all terms by -0.4 the coefficient of the squared term: Divide each side by '-0.4'. -2500 + 105x + x2 = 0 Move the constant term to the right: Add '2500' to each side of the equation. -2500 + 105x + 2500 + x2 = 0 + 2500 Reorder the terms: -2500 + 2500 + 105x + x2 = 0 + 2500 Combine like terms: -2500 + 2500 = 0 0 + 105x + x2 = 0 + 2500 105x + x2 = 0 + 2500 Combine like terms: 0 + 2500 = 2500 105x + x2 = 2500 The x term is 105x. Take half its coefficient (52.5). Square it (2756.25) and add it to both sides. Add '2756.25' to each side of the equation. 105x + 2756.25 + x2 = 2500 + 2756.25 Reorder the terms: 2756.25 + 105x + x2 = 2500 + 2756.25 Combine like terms: 2500 + 2756.25 = 5256.25 2756.25 + 105x + x2 = 5256.25 Factor a perfect square on the left side: (x + 52.5)(x + 52.5) = 5256.25 Calculate the square root of the right side: 72.5 Break this problem into two subproblems by setting (x + 52.5) equal to 72.5 and -72.5.Subproblem 1
x + 52.5 = 72.5 Simplifying x + 52.5 = 72.5 Reorder the terms: 52.5 + x = 72.5 Solving 52.5 + x = 72.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-52.5' to each side of the equation. 52.5 + -52.5 + x = 72.5 + -52.5 Combine like terms: 52.5 + -52.5 = 0.0 0.0 + x = 72.5 + -52.5 x = 72.5 + -52.5 Combine like terms: 72.5 + -52.5 = 20 x = 20 Simplifying x = 20Subproblem 2
x + 52.5 = -72.5 Simplifying x + 52.5 = -72.5 Reorder the terms: 52.5 + x = -72.5 Solving 52.5 + x = -72.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-52.5' to each side of the equation. 52.5 + -52.5 + x = -72.5 + -52.5 Combine like terms: 52.5 + -52.5 = 0.0 0.0 + x = -72.5 + -52.5 x = -72.5 + -52.5 Combine like terms: -72.5 + -52.5 = -125 x = -125 Simplifying x = -125Solution
The solution to the problem is based on the solutions from the subproblems. x = {20, -125}
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