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Simplifying (5x + -6)(9x + -2) = x Reorder the terms: (-6 + 5x)(9x + -2) = x Reorder the terms: (-6 + 5x)(-2 + 9x) = x Multiply (-6 + 5x) * (-2 + 9x) (-6(-2 + 9x) + 5x * (-2 + 9x)) = x ((-2 * -6 + 9x * -6) + 5x * (-2 + 9x)) = x ((12 + -54x) + 5x * (-2 + 9x)) = x (12 + -54x + (-2 * 5x + 9x * 5x)) = x (12 + -54x + (-10x + 45x2)) = x Combine like terms: -54x + -10x = -64x (12 + -64x + 45x2) = x Solving 12 + -64x + 45x2 = x Solving for variable 'x'. Reorder the terms: 12 + -64x + -1x + 45x2 = x + -1x Combine like terms: -64x + -1x = -65x 12 + -65x + 45x2 = x + -1x Combine like terms: x + -1x = 0 12 + -65x + 45x2 = 0 Begin completing the square. Divide all terms by 45 the coefficient of the squared term: Divide each side by '45'. 0.2666666667 + -1.444444444x + x2 = 0 Move the constant term to the right: Add '-0.2666666667' to each side of the equation. 0.2666666667 + -1.444444444x + -0.2666666667 + x2 = 0 + -0.2666666667 Reorder the terms: 0.2666666667 + -0.2666666667 + -1.444444444x + x2 = 0 + -0.2666666667 Combine like terms: 0.2666666667 + -0.2666666667 = 0.0000000000 0.0000000000 + -1.444444444x + x2 = 0 + -0.2666666667 -1.444444444x + x2 = 0 + -0.2666666667 Combine like terms: 0 + -0.2666666667 = -0.2666666667 -1.444444444x + x2 = -0.2666666667 The x term is -1.444444444x. Take half its coefficient (-0.722222222). Square it (0.5216049380) and add it to both sides. Add '0.5216049380' to each side of the equation. -1.444444444x + 0.5216049380 + x2 = -0.2666666667 + 0.5216049380 Reorder the terms: 0.5216049380 + -1.444444444x + x2 = -0.2666666667 + 0.5216049380 Combine like terms: -0.2666666667 + 0.5216049380 = 0.2549382713 0.5216049380 + -1.444444444x + x2 = 0.2549382713 Factor a perfect square on the left side: (x + -0.722222222)(x + -0.722222222) = 0.2549382713 Calculate the square root of the right side: 0.504914123 Break this problem into two subproblems by setting (x + -0.722222222) equal to 0.504914123 and -0.504914123.Subproblem 1
x + -0.722222222 = 0.504914123 Simplifying x + -0.722222222 = 0.504914123 Reorder the terms: -0.722222222 + x = 0.504914123 Solving -0.722222222 + x = 0.504914123 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.722222222' to each side of the equation. -0.722222222 + 0.722222222 + x = 0.504914123 + 0.722222222 Combine like terms: -0.722222222 + 0.722222222 = 0.000000000 0.000000000 + x = 0.504914123 + 0.722222222 x = 0.504914123 + 0.722222222 Combine like terms: 0.504914123 + 0.722222222 = 1.227136345 x = 1.227136345 Simplifying x = 1.227136345Subproblem 2
x + -0.722222222 = -0.504914123 Simplifying x + -0.722222222 = -0.504914123 Reorder the terms: -0.722222222 + x = -0.504914123 Solving -0.722222222 + x = -0.504914123 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.722222222' to each side of the equation. -0.722222222 + 0.722222222 + x = -0.504914123 + 0.722222222 Combine like terms: -0.722222222 + 0.722222222 = 0.000000000 0.000000000 + x = -0.504914123 + 0.722222222 x = -0.504914123 + 0.722222222 Combine like terms: -0.504914123 + 0.722222222 = 0.217308099 x = 0.217308099 Simplifying x = 0.217308099Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.227136345, 0.217308099}
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