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Simplifying 2x(10 + x) = 1(x + 42) (10 * 2x + x * 2x) = 1(x + 42) (20x + 2x2) = 1(x + 42) Reorder the terms: 20x + 2x2 = 1(42 + x) 20x + 2x2 = (42 * 1 + x * 1) 20x + 2x2 = (42 + 1x) Solving 20x + 2x2 = 42 + 1x Solving for variable 'x'. Reorder the terms: -42 + 20x + -1x + 2x2 = 42 + 1x + -42 + -1x Combine like terms: 20x + -1x = 19x -42 + 19x + 2x2 = 42 + 1x + -42 + -1x Reorder the terms: -42 + 19x + 2x2 = 42 + -42 + 1x + -1x Combine like terms: 42 + -42 = 0 -42 + 19x + 2x2 = 0 + 1x + -1x -42 + 19x + 2x2 = 1x + -1x Combine like terms: 1x + -1x = 0 -42 + 19x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -21 + 9.5x + x2 = 0 Move the constant term to the right: Add '21' to each side of the equation. -21 + 9.5x + 21 + x2 = 0 + 21 Reorder the terms: -21 + 21 + 9.5x + x2 = 0 + 21 Combine like terms: -21 + 21 = 0 0 + 9.5x + x2 = 0 + 21 9.5x + x2 = 0 + 21 Combine like terms: 0 + 21 = 21 9.5x + x2 = 21 The x term is 9.5x. Take half its coefficient (4.75). Square it (22.5625) and add it to both sides. Add '22.5625' to each side of the equation. 9.5x + 22.5625 + x2 = 21 + 22.5625 Reorder the terms: 22.5625 + 9.5x + x2 = 21 + 22.5625 Combine like terms: 21 + 22.5625 = 43.5625 22.5625 + 9.5x + x2 = 43.5625 Factor a perfect square on the left side: (x + 4.75)(x + 4.75) = 43.5625 Calculate the square root of the right side: 6.600189391 Break this problem into two subproblems by setting (x + 4.75) equal to 6.600189391 and -6.600189391.Subproblem 1
x + 4.75 = 6.600189391 Simplifying x + 4.75 = 6.600189391 Reorder the terms: 4.75 + x = 6.600189391 Solving 4.75 + x = 6.600189391 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.75' to each side of the equation. 4.75 + -4.75 + x = 6.600189391 + -4.75 Combine like terms: 4.75 + -4.75 = 0.00 0.00 + x = 6.600189391 + -4.75 x = 6.600189391 + -4.75 Combine like terms: 6.600189391 + -4.75 = 1.850189391 x = 1.850189391 Simplifying x = 1.850189391Subproblem 2
x + 4.75 = -6.600189391 Simplifying x + 4.75 = -6.600189391 Reorder the terms: 4.75 + x = -6.600189391 Solving 4.75 + x = -6.600189391 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.75' to each side of the equation. 4.75 + -4.75 + x = -6.600189391 + -4.75 Combine like terms: 4.75 + -4.75 = 0.00 0.00 + x = -6.600189391 + -4.75 x = -6.600189391 + -4.75 Combine like terms: -6.600189391 + -4.75 = -11.350189391 x = -11.350189391 Simplifying x = -11.350189391Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.850189391, -11.350189391}
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