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Simplifying 2x(3x + 1) = 5x + 7 Reorder the terms: 2x(1 + 3x) = 5x + 7 (1 * 2x + 3x * 2x) = 5x + 7 (2x + 6x2) = 5x + 7 Reorder the terms: 2x + 6x2 = 7 + 5x Solving 2x + 6x2 = 7 + 5x Solving for variable 'x'. Reorder the terms: -7 + 2x + -5x + 6x2 = 7 + 5x + -7 + -5x Combine like terms: 2x + -5x = -3x -7 + -3x + 6x2 = 7 + 5x + -7 + -5x Reorder the terms: -7 + -3x + 6x2 = 7 + -7 + 5x + -5x Combine like terms: 7 + -7 = 0 -7 + -3x + 6x2 = 0 + 5x + -5x -7 + -3x + 6x2 = 5x + -5x Combine like terms: 5x + -5x = 0 -7 + -3x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -1.166666667 + -0.5x + x2 = 0 Move the constant term to the right: Add '1.166666667' to each side of the equation. -1.166666667 + -0.5x + 1.166666667 + x2 = 0 + 1.166666667 Reorder the terms: -1.166666667 + 1.166666667 + -0.5x + x2 = 0 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + -0.5x + x2 = 0 + 1.166666667 -0.5x + x2 = 0 + 1.166666667 Combine like terms: 0 + 1.166666667 = 1.166666667 -0.5x + x2 = 1.166666667 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 = 1.166666667 + 0.0625 Reorder the terms: 0.0625 + -0.5x + x2 = 1.166666667 + 0.0625 Combine like terms: 1.166666667 + 0.0625 = 1.229166667 0.0625 + -0.5x + x2 = 1.229166667 Factor a perfect square on the left side: (x + -0.25)(x + -0.25) = 1.229166667 Calculate the square root of the right side: 1.108677891 Break this problem into two subproblems by setting (x + -0.25) equal to 1.108677891 and -1.108677891.Subproblem 1
x + -0.25 = 1.108677891 Simplifying x + -0.25 = 1.108677891 Reorder the terms: -0.25 + x = 1.108677891 Solving -0.25 + x = 1.108677891 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 = 1.108677891 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = 1.108677891 + 0.25 x = 1.108677891 + 0.25 Combine like terms: 1.108677891 + 0.25 = 1.358677891 x = 1.358677891 Simplifying x = 1.358677891Subproblem 2
x + -0.25 = -1.108677891 Simplifying x + -0.25 = -1.108677891 Reorder the terms: -0.25 + x = -1.108677891 Solving -0.25 + x = -1.108677891 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 = -1.108677891 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = -1.108677891 + 0.25 x = -1.108677891 + 0.25 Combine like terms: -1.108677891 + 0.25 = -0.858677891 x = -0.858677891 Simplifying x = -0.858677891Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.358677891, -0.858677891}
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