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Simplifying 0.20(65 + -1x) = x2 (65 * 0.20 + -1x * 0.20) = x2 (13 + -0.2x) = x2 Solving 13 + -0.2x = x2 Solving for variable 'x'. Combine like terms: x2 + -1x2 = 0 13 + -0.2x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -13 + 0.2x + x2 = 0 Move the constant term to the right: Add '13' to each side of the equation. -13 + 0.2x + 13 + x2 = 0 + 13 Reorder the terms: -13 + 13 + 0.2x + x2 = 0 + 13 Combine like terms: -13 + 13 = 0 0 + 0.2x + x2 = 0 + 13 0.2x + x2 = 0 + 13 Combine like terms: 0 + 13 = 13 0.2x + x2 = 13 The x term is 0.2x. Take half its coefficient (0.1). Square it (0.01) and add it to both sides. Add '0.01' to each side of the equation. 0.2x + 0.01 + x2 = 13 + 0.01 Reorder the terms: 0.01 + 0.2x + x2 = 13 + 0.01 Combine like terms: 13 + 0.01 = 13.01 0.01 + 0.2x + x2 = 13.01 Factor a perfect square on the left side: (x + 0.1)(x + 0.1) = 13.01 Calculate the square root of the right side: 3.606937759 Break this problem into two subproblems by setting (x + 0.1) equal to 3.606937759 and -3.606937759.Subproblem 1
x + 0.1 = 3.606937759 Simplifying x + 0.1 = 3.606937759 Reorder the terms: 0.1 + x = 3.606937759 Solving 0.1 + x = 3.606937759 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1' to each side of the equation. 0.1 + -0.1 + x = 3.606937759 + -0.1 Combine like terms: 0.1 + -0.1 = 0.0 0.0 + x = 3.606937759 + -0.1 x = 3.606937759 + -0.1 Combine like terms: 3.606937759 + -0.1 = 3.506937759 x = 3.506937759 Simplifying x = 3.506937759Subproblem 2
x + 0.1 = -3.606937759 Simplifying x + 0.1 = -3.606937759 Reorder the terms: 0.1 + x = -3.606937759 Solving 0.1 + x = -3.606937759 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1' to each side of the equation. 0.1 + -0.1 + x = -3.606937759 + -0.1 Combine like terms: 0.1 + -0.1 = 0.0 0.0 + x = -3.606937759 + -0.1 x = -3.606937759 + -0.1 Combine like terms: -3.606937759 + -0.1 = -3.706937759 x = -3.706937759 Simplifying x = -3.706937759Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.506937759, -3.706937759}
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