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Simplifying x(8x + 13) = 3x2 + 8x + 54 Reorder the terms: x(13 + 8x) = 3x2 + 8x + 54 (13 * x + 8x * x) = 3x2 + 8x + 54 (13x + 8x2) = 3x2 + 8x + 54 Reorder the terms: 13x + 8x2 = 54 + 8x + 3x2 Solving 13x + 8x2 = 54 + 8x + 3x2 Solving for variable 'x'. Reorder the terms: -54 + 13x + -8x + 8x2 + -3x2 = 54 + 8x + 3x2 + -54 + -8x + -3x2 Combine like terms: 13x + -8x = 5x -54 + 5x + 8x2 + -3x2 = 54 + 8x + 3x2 + -54 + -8x + -3x2 Combine like terms: 8x2 + -3x2 = 5x2 -54 + 5x + 5x2 = 54 + 8x + 3x2 + -54 + -8x + -3x2 Reorder the terms: -54 + 5x + 5x2 = 54 + -54 + 8x + -8x + 3x2 + -3x2 Combine like terms: 54 + -54 = 0 -54 + 5x + 5x2 = 0 + 8x + -8x + 3x2 + -3x2 -54 + 5x + 5x2 = 8x + -8x + 3x2 + -3x2 Combine like terms: 8x + -8x = 0 -54 + 5x + 5x2 = 0 + 3x2 + -3x2 -54 + 5x + 5x2 = 3x2 + -3x2 Combine like terms: 3x2 + -3x2 = 0 -54 + 5x + 5x2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -10.8 + x + x2 = 0 Move the constant term to the right: Add '10.8' to each side of the equation. -10.8 + x + 10.8 + x2 = 0 + 10.8 Reorder the terms: -10.8 + 10.8 + x + x2 = 0 + 10.8 Combine like terms: -10.8 + 10.8 = 0.0 0.0 + x + x2 = 0 + 10.8 x + x2 = 0 + 10.8 Combine like terms: 0 + 10.8 = 10.8 x + x2 = 10.8 The x term is x. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + x2 = 10.8 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + x2 = 10.8 + 0.25 Combine like terms: 10.8 + 0.25 = 11.05 1.25 + x2 = 11.05 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 11.05 Calculate the square root of the right side: 3.324154028 Break this problem into two subproblems by setting (x + 0.5) equal to 3.324154028 and -3.324154028.Subproblem 1
x + 0.5 = 3.324154028 Simplifying x + 0.5 = 3.324154028 Reorder the terms: 0.5 + x = 3.324154028 Solving 0.5 + x = 3.324154028 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 3.324154028 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 3.324154028 + -0.5 x = 3.324154028 + -0.5 Combine like terms: 3.324154028 + -0.5 = 2.824154028 x = 2.824154028 Simplifying x = 2.824154028Subproblem 2
x + 0.5 = -3.324154028 Simplifying x + 0.5 = -3.324154028 Reorder the terms: 0.5 + x = -3.324154028 Solving 0.5 + x = -3.324154028 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -3.324154028 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -3.324154028 + -0.5 x = -3.324154028 + -0.5 Combine like terms: -3.324154028 + -0.5 = -3.824154028 x = -3.824154028 Simplifying x = -3.824154028Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.824154028, -3.824154028}
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