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Simplifying x(2x + 1) = 42 Reorder the terms: x(1 + 2x) = 42 (1 * x + 2x * x) = 42 (1x + 2x2) = 42 Solving 1x + 2x2 = 42 Solving for variable 'x'. Reorder the terms: -42 + 1x + 2x2 = 42 + -42 Combine like terms: 42 + -42 = 0 -42 + 1x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -21 + 0.5x + x2 = 0 Move the constant term to the right: Add '21' to each side of the equation. -21 + 0.5x + 21 + x2 = 0 + 21 Reorder the terms: -21 + 21 + 0.5x + x2 = 0 + 21 Combine like terms: -21 + 21 = 0 0 + 0.5x + x2 = 0 + 21 0.5x + x2 = 0 + 21 Combine like terms: 0 + 21 = 21 0.5x + x2 = 21 The x term is 0.5x. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5x + 0.0625 + x2 = 21 + 0.0625 Reorder the terms: 0.0625 + 0.5x + x2 = 21 + 0.0625 Combine like terms: 21 + 0.0625 = 21.0625 0.0625 + 0.5x + x2 = 21.0625 Factor a perfect square on the left side: (x + 0.25)(x + 0.25) = 21.0625 Calculate the square root of the right side: 4.589389938 Break this problem into two subproblems by setting (x + 0.25) equal to 4.589389938 and -4.589389938.Subproblem 1
x + 0.25 = 4.589389938 Simplifying x + 0.25 = 4.589389938 Reorder the terms: 0.25 + x = 4.589389938 Solving 0.25 + x = 4.589389938 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = 4.589389938 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 4.589389938 + -0.25 x = 4.589389938 + -0.25 Combine like terms: 4.589389938 + -0.25 = 4.339389938 x = 4.339389938 Simplifying x = 4.339389938Subproblem 2
x + 0.25 = -4.589389938 Simplifying x + 0.25 = -4.589389938 Reorder the terms: 0.25 + x = -4.589389938 Solving 0.25 + x = -4.589389938 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = -4.589389938 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -4.589389938 + -0.25 x = -4.589389938 + -0.25 Combine like terms: -4.589389938 + -0.25 = -4.839389938 x = -4.839389938 Simplifying x = -4.839389938Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.339389938, -4.839389938}
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