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Simplifying (14x + 6)(15 + -12x) = 1 Reorder the terms: (6 + 14x)(15 + -12x) = 1 Multiply (6 + 14x) * (15 + -12x) (6(15 + -12x) + 14x * (15 + -12x)) = 1 ((15 * 6 + -12x * 6) + 14x * (15 + -12x)) = 1 ((90 + -72x) + 14x * (15 + -12x)) = 1 (90 + -72x + (15 * 14x + -12x * 14x)) = 1 (90 + -72x + (210x + -168x2)) = 1 Combine like terms: -72x + 210x = 138x (90 + 138x + -168x2) = 1 Solving 90 + 138x + -168x2 = 1 Solving for variable 'x'. Reorder the terms: 90 + -1 + 138x + -168x2 = 1 + -1 Combine like terms: 90 + -1 = 89 89 + 138x + -168x2 = 1 + -1 Combine like terms: 1 + -1 = 0 89 + 138x + -168x2 = 0 Begin completing the square. Divide all terms by -168 the coefficient of the squared term: Divide each side by '-168'. -0.5297619048 + -0.8214285714x + x2 = 0 Move the constant term to the right: Add '0.5297619048' to each side of the equation. -0.5297619048 + -0.8214285714x + 0.5297619048 + x2 = 0 + 0.5297619048 Reorder the terms: -0.5297619048 + 0.5297619048 + -0.8214285714x + x2 = 0 + 0.5297619048 Combine like terms: -0.5297619048 + 0.5297619048 = 0.0000000000 0.0000000000 + -0.8214285714x + x2 = 0 + 0.5297619048 -0.8214285714x + x2 = 0 + 0.5297619048 Combine like terms: 0 + 0.5297619048 = 0.5297619048 -0.8214285714x + x2 = 0.5297619048 The x term is -0.8214285714x. Take half its coefficient (-0.4107142857). Square it (0.1686862245) and add it to both sides. Add '0.1686862245' to each side of the equation. -0.8214285714x + 0.1686862245 + x2 = 0.5297619048 + 0.1686862245 Reorder the terms: 0.1686862245 + -0.8214285714x + x2 = 0.5297619048 + 0.1686862245 Combine like terms: 0.5297619048 + 0.1686862245 = 0.6984481293 0.1686862245 + -0.8214285714x + x2 = 0.6984481293 Factor a perfect square on the left side: (x + -0.4107142857)(x + -0.4107142857) = 0.6984481293 Calculate the square root of the right side: 0.835732092 Break this problem into two subproblems by setting (x + -0.4107142857) equal to 0.835732092 and -0.835732092.Subproblem 1
x + -0.4107142857 = 0.835732092 Simplifying x + -0.4107142857 = 0.835732092 Reorder the terms: -0.4107142857 + x = 0.835732092 Solving -0.4107142857 + x = 0.835732092 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4107142857' to each side of the equation. -0.4107142857 + 0.4107142857 + x = 0.835732092 + 0.4107142857 Combine like terms: -0.4107142857 + 0.4107142857 = 0.0000000000 0.0000000000 + x = 0.835732092 + 0.4107142857 x = 0.835732092 + 0.4107142857 Combine like terms: 0.835732092 + 0.4107142857 = 1.2464463777 x = 1.2464463777 Simplifying x = 1.2464463777Subproblem 2
x + -0.4107142857 = -0.835732092 Simplifying x + -0.4107142857 = -0.835732092 Reorder the terms: -0.4107142857 + x = -0.835732092 Solving -0.4107142857 + x = -0.835732092 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4107142857' to each side of the equation. -0.4107142857 + 0.4107142857 + x = -0.835732092 + 0.4107142857 Combine like terms: -0.4107142857 + 0.4107142857 = 0.0000000000 0.0000000000 + x = -0.835732092 + 0.4107142857 x = -0.835732092 + 0.4107142857 Combine like terms: -0.835732092 + 0.4107142857 = -0.4250178063 x = -0.4250178063 Simplifying x = -0.4250178063Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.2464463777, -0.4250178063}
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