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Simplifying 0.65 = x(x + 25) Reorder the terms: 0.65 = x(25 + x) 0.65 = (25 * x + x * x) 0.65 = (25x + x2) Solving 0.65 = 25x + x2 Solving for variable 'x'. Reorder the terms: 0.65 + -25x + -1x2 = 25x + -25x + x2 + -1x2 Combine like terms: 25x + -25x = 0 0.65 + -25x + -1x2 = 0 + x2 + -1x2 0.65 + -25x + -1x2 = x2 + -1x2 Combine like terms: x2 + -1x2 = 0 0.65 + -25x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -0.65 + 25x + x2 = 0 Move the constant term to the right: Add '0.65' to each side of the equation. -0.65 + 25x + 0.65 + x2 = 0 + 0.65 Reorder the terms: -0.65 + 0.65 + 25x + x2 = 0 + 0.65 Combine like terms: -0.65 + 0.65 = 0.00 0.00 + 25x + x2 = 0 + 0.65 25x + x2 = 0 + 0.65 Combine like terms: 0 + 0.65 = 0.65 25x + x2 = 0.65 The x term is 25x. Take half its coefficient (12.5). Square it (156.25) and add it to both sides. Add '156.25' to each side of the equation. 25x + 156.25 + x2 = 0.65 + 156.25 Reorder the terms: 156.25 + 25x + x2 = 0.65 + 156.25 Combine like terms: 0.65 + 156.25 = 156.9 156.25 + 25x + x2 = 156.9 Factor a perfect square on the left side: (x + 12.5)(x + 12.5) = 156.9 Calculate the square root of the right side: 12.525973016 Break this problem into two subproblems by setting (x + 12.5) equal to 12.525973016 and -12.525973016.Subproblem 1
x + 12.5 = 12.525973016 Simplifying x + 12.5 = 12.525973016 Reorder the terms: 12.5 + x = 12.525973016 Solving 12.5 + x = 12.525973016 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-12.5' to each side of the equation. 12.5 + -12.5 + x = 12.525973016 + -12.5 Combine like terms: 12.5 + -12.5 = 0.0 0.0 + x = 12.525973016 + -12.5 x = 12.525973016 + -12.5 Combine like terms: 12.525973016 + -12.5 = 0.025973016 x = 0.025973016 Simplifying x = 0.025973016Subproblem 2
x + 12.5 = -12.525973016 Simplifying x + 12.5 = -12.525973016 Reorder the terms: 12.5 + x = -12.525973016 Solving 12.5 + x = -12.525973016 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-12.5' to each side of the equation. 12.5 + -12.5 + x = -12.525973016 + -12.5 Combine like terms: 12.5 + -12.5 = 0.0 0.0 + x = -12.525973016 + -12.5 x = -12.525973016 + -12.5 Combine like terms: -12.525973016 + -12.5 = -25.025973016 x = -25.025973016 Simplifying x = -25.025973016Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.025973016, -25.025973016}
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