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Simplifying 8x(2x + 3) = 4x + 1 Reorder the terms: 8x(3 + 2x) = 4x + 1 (3 * 8x + 2x * 8x) = 4x + 1 (24x + 16x2) = 4x + 1 Reorder the terms: 24x + 16x2 = 1 + 4x Solving 24x + 16x2 = 1 + 4x Solving for variable 'x'. Reorder the terms: -1 + 24x + -4x + 16x2 = 1 + 4x + -1 + -4x Combine like terms: 24x + -4x = 20x -1 + 20x + 16x2 = 1 + 4x + -1 + -4x Reorder the terms: -1 + 20x + 16x2 = 1 + -1 + 4x + -4x Combine like terms: 1 + -1 = 0 -1 + 20x + 16x2 = 0 + 4x + -4x -1 + 20x + 16x2 = 4x + -4x Combine like terms: 4x + -4x = 0 -1 + 20x + 16x2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.0625 + 1.25x + x2 = 0 Move the constant term to the right: Add '0.0625' to each side of the equation. -0.0625 + 1.25x + 0.0625 + x2 = 0 + 0.0625 Reorder the terms: -0.0625 + 0.0625 + 1.25x + x2 = 0 + 0.0625 Combine like terms: -0.0625 + 0.0625 = 0.0000 0.0000 + 1.25x + x2 = 0 + 0.0625 1.25x + x2 = 0 + 0.0625 Combine like terms: 0 + 0.0625 = 0.0625 1.25x + x2 = 0.0625 The x term is 1.25x. Take half its coefficient (0.625). Square it (0.390625) and add it to both sides. Add '0.390625' to each side of the equation. 1.25x + 0.390625 + x2 = 0.0625 + 0.390625 Reorder the terms: 0.390625 + 1.25x + x2 = 0.0625 + 0.390625 Combine like terms: 0.0625 + 0.390625 = 0.453125 0.390625 + 1.25x + x2 = 0.453125 Factor a perfect square on the left side: (x + 0.625)(x + 0.625) = 0.453125 Calculate the square root of the right side: 0.673145601 Break this problem into two subproblems by setting (x + 0.625) equal to 0.673145601 and -0.673145601.Subproblem 1
x + 0.625 = 0.673145601 Simplifying x + 0.625 = 0.673145601 Reorder the terms: 0.625 + x = 0.673145601 Solving 0.625 + x = 0.673145601 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + x = 0.673145601 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + x = 0.673145601 + -0.625 x = 0.673145601 + -0.625 Combine like terms: 0.673145601 + -0.625 = 0.048145601 x = 0.048145601 Simplifying x = 0.048145601Subproblem 2
x + 0.625 = -0.673145601 Simplifying x + 0.625 = -0.673145601 Reorder the terms: 0.625 + x = -0.673145601 Solving 0.625 + x = -0.673145601 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + x = -0.673145601 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + x = -0.673145601 + -0.625 x = -0.673145601 + -0.625 Combine like terms: -0.673145601 + -0.625 = -1.298145601 x = -1.298145601 Simplifying x = -1.298145601Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.048145601, -1.298145601}
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