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Simplifying x(5x + 1) = 2x2 + 4x + 28 Reorder the terms: x(1 + 5x) = 2x2 + 4x + 28 (1 * x + 5x * x) = 2x2 + 4x + 28 (1x + 5x2) = 2x2 + 4x + 28 Reorder the terms: 1x + 5x2 = 28 + 4x + 2x2 Solving 1x + 5x2 = 28 + 4x + 2x2 Solving for variable 'x'. Reorder the terms: -28 + 1x + -4x + 5x2 + -2x2 = 28 + 4x + 2x2 + -28 + -4x + -2x2 Combine like terms: 1x + -4x = -3x -28 + -3x + 5x2 + -2x2 = 28 + 4x + 2x2 + -28 + -4x + -2x2 Combine like terms: 5x2 + -2x2 = 3x2 -28 + -3x + 3x2 = 28 + 4x + 2x2 + -28 + -4x + -2x2 Reorder the terms: -28 + -3x + 3x2 = 28 + -28 + 4x + -4x + 2x2 + -2x2 Combine like terms: 28 + -28 = 0 -28 + -3x + 3x2 = 0 + 4x + -4x + 2x2 + -2x2 -28 + -3x + 3x2 = 4x + -4x + 2x2 + -2x2 Combine like terms: 4x + -4x = 0 -28 + -3x + 3x2 = 0 + 2x2 + -2x2 -28 + -3x + 3x2 = 2x2 + -2x2 Combine like terms: 2x2 + -2x2 = 0 -28 + -3x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -9.333333333 + -1x + x2 = 0 Move the constant term to the right: Add '9.333333333' to each side of the equation. -9.333333333 + -1x + 9.333333333 + x2 = 0 + 9.333333333 Reorder the terms: -9.333333333 + 9.333333333 + -1x + x2 = 0 + 9.333333333 Combine like terms: -9.333333333 + 9.333333333 = 0.000000000 0.000000000 + -1x + x2 = 0 + 9.333333333 -1x + x2 = 0 + 9.333333333 Combine like terms: 0 + 9.333333333 = 9.333333333 -1x + x2 = 9.333333333 The x term is -1x. 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. -1x + 0.25 + x2 = 9.333333333 + 0.25 Reorder the terms: 0.25 + -1x + x2 = 9.333333333 + 0.25 Combine like terms: 9.333333333 + 0.25 = 9.583333333 0.25 + -1x + x2 = 9.583333333 Factor a perfect square on the left side: (x + -0.5)(x + -0.5) = 9.583333333 Calculate the square root of the right side: 3.095695937 Break this problem into two subproblems by setting (x + -0.5) equal to 3.095695937 and -3.095695937.Subproblem 1
x + -0.5 = 3.095695937 Simplifying x + -0.5 = 3.095695937 Reorder the terms: -0.5 + x = 3.095695937 Solving -0.5 + x = 3.095695937 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 = 3.095695937 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 3.095695937 + 0.5 x = 3.095695937 + 0.5 Combine like terms: 3.095695937 + 0.5 = 3.595695937 x = 3.595695937 Simplifying x = 3.595695937Subproblem 2
x + -0.5 = -3.095695937 Simplifying x + -0.5 = -3.095695937 Reorder the terms: -0.5 + x = -3.095695937 Solving -0.5 + x = -3.095695937 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 = -3.095695937 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -3.095695937 + 0.5 x = -3.095695937 + 0.5 Combine like terms: -3.095695937 + 0.5 = -2.595695937 x = -2.595695937 Simplifying x = -2.595695937Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.595695937, -2.595695937}
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