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Simplifying 5(2x + -1)(3x + 1) = 1 Reorder the terms: 5(-1 + 2x)(3x + 1) = 1 Reorder the terms: 5(-1 + 2x)(1 + 3x) = 1 Multiply (-1 + 2x) * (1 + 3x) 5(-1(1 + 3x) + 2x * (1 + 3x)) = 1 5((1 * -1 + 3x * -1) + 2x * (1 + 3x)) = 1 5((-1 + -3x) + 2x * (1 + 3x)) = 1 5(-1 + -3x + (1 * 2x + 3x * 2x)) = 1 5(-1 + -3x + (2x + 6x2)) = 1 Combine like terms: -3x + 2x = -1x 5(-1 + -1x + 6x2) = 1 (-1 * 5 + -1x * 5 + 6x2 * 5) = 1 (-5 + -5x + 30x2) = 1 Solving -5 + -5x + 30x2 = 1 Solving for variable 'x'. Reorder the terms: -5 + -1 + -5x + 30x2 = 1 + -1 Combine like terms: -5 + -1 = -6 -6 + -5x + 30x2 = 1 + -1 Combine like terms: 1 + -1 = 0 -6 + -5x + 30x2 = 0 Begin completing the square. Divide all terms by 30 the coefficient of the squared term: Divide each side by '30'. -0.2 + -0.1666666667x + x2 = 0.0 Move the constant term to the right: Add '0.2' to each side of the equation. -0.2 + -0.1666666667x + 0.2 + x2 = 0.0 + 0.2 Reorder the terms: -0.2 + 0.2 + -0.1666666667x + x2 = 0.0 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + -0.1666666667x + x2 = 0.0 + 0.2 -0.1666666667x + x2 = 0.0 + 0.2 Combine like terms: 0.0 + 0.2 = 0.2 -0.1666666667x + x2 = 0.2 The x term is -0.1666666667x. Take half its coefficient (-0.08333333335). Square it (0.006944444447) and add it to both sides. Add '0.006944444447' to each side of the equation. -0.1666666667x + 0.006944444447 + x2 = 0.2 + 0.006944444447 Reorder the terms: 0.006944444447 + -0.1666666667x + x2 = 0.2 + 0.006944444447 Combine like terms: 0.2 + 0.006944444447 = 0.206944444447 0.006944444447 + -0.1666666667x + x2 = 0.206944444447 Factor a perfect square on the left side: (x + -0.08333333335)(x + -0.08333333335) = 0.206944444447 Calculate the square root of the right side: 0.454911469 Break this problem into two subproblems by setting (x + -0.08333333335) equal to 0.454911469 and -0.454911469.Subproblem 1
x + -0.08333333335 = 0.454911469 Simplifying x + -0.08333333335 = 0.454911469 Reorder the terms: -0.08333333335 + x = 0.454911469 Solving -0.08333333335 + x = 0.454911469 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.08333333335' to each side of the equation. -0.08333333335 + 0.08333333335 + x = 0.454911469 + 0.08333333335 Combine like terms: -0.08333333335 + 0.08333333335 = 0.00000000000 0.00000000000 + x = 0.454911469 + 0.08333333335 x = 0.454911469 + 0.08333333335 Combine like terms: 0.454911469 + 0.08333333335 = 0.53824480235 x = 0.53824480235 Simplifying x = 0.53824480235Subproblem 2
x + -0.08333333335 = -0.454911469 Simplifying x + -0.08333333335 = -0.454911469 Reorder the terms: -0.08333333335 + x = -0.454911469 Solving -0.08333333335 + x = -0.454911469 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.08333333335' to each side of the equation. -0.08333333335 + 0.08333333335 + x = -0.454911469 + 0.08333333335 Combine like terms: -0.08333333335 + 0.08333333335 = 0.00000000000 0.00000000000 + x = -0.454911469 + 0.08333333335 x = -0.454911469 + 0.08333333335 Combine like terms: -0.454911469 + 0.08333333335 = -0.37157813565 x = -0.37157813565 Simplifying x = -0.37157813565Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.53824480235, -0.37157813565}
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