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Simplifying (8x + 2)(9x + 3) + x = 180 Reorder the terms: (2 + 8x)(9x + 3) + x = 180 Reorder the terms: (2 + 8x)(3 + 9x) + x = 180 Multiply (2 + 8x) * (3 + 9x) (2(3 + 9x) + 8x * (3 + 9x)) + x = 180 ((3 * 2 + 9x * 2) + 8x * (3 + 9x)) + x = 180 ((6 + 18x) + 8x * (3 + 9x)) + x = 180 (6 + 18x + (3 * 8x + 9x * 8x)) + x = 180 (6 + 18x + (24x + 72x2)) + x = 180 Combine like terms: 18x + 24x = 42x (6 + 42x + 72x2) + x = 180 Reorder the terms: 6 + 42x + x + 72x2 = 180 Combine like terms: 42x + x = 43x 6 + 43x + 72x2 = 180 Solving 6 + 43x + 72x2 = 180 Solving for variable 'x'. Reorder the terms: 6 + -180 + 43x + 72x2 = 180 + -180 Combine like terms: 6 + -180 = -174 -174 + 43x + 72x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -174 + 43x + 72x2 = 0 Begin completing the square. Divide all terms by 72 the coefficient of the squared term: Divide each side by '72'. -2.416666667 + 0.5972222222x + x2 = 0 Move the constant term to the right: Add '2.416666667' to each side of the equation. -2.416666667 + 0.5972222222x + 2.416666667 + x2 = 0 + 2.416666667 Reorder the terms: -2.416666667 + 2.416666667 + 0.5972222222x + x2 = 0 + 2.416666667 Combine like terms: -2.416666667 + 2.416666667 = 0.000000000 0.000000000 + 0.5972222222x + x2 = 0 + 2.416666667 0.5972222222x + x2 = 0 + 2.416666667 Combine like terms: 0 + 2.416666667 = 2.416666667 0.5972222222x + x2 = 2.416666667 The x term is 0.5972222222x. Take half its coefficient (0.2986111111). Square it (0.08916859567) and add it to both sides. Add '0.08916859567' to each side of the equation. 0.5972222222x + 0.08916859567 + x2 = 2.416666667 + 0.08916859567 Reorder the terms: 0.08916859567 + 0.5972222222x + x2 = 2.416666667 + 0.08916859567 Combine like terms: 2.416666667 + 0.08916859567 = 2.50583526267 0.08916859567 + 0.5972222222x + x2 = 2.50583526267 Factor a perfect square on the left side: (x + 0.2986111111)(x + 0.2986111111) = 2.50583526267 Calculate the square root of the right side: 1.582983027 Break this problem into two subproblems by setting (x + 0.2986111111) equal to 1.582983027 and -1.582983027.Subproblem 1
x + 0.2986111111 = 1.582983027 Simplifying x + 0.2986111111 = 1.582983027 Reorder the terms: 0.2986111111 + x = 1.582983027 Solving 0.2986111111 + x = 1.582983027 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.2986111111' to each side of the equation. 0.2986111111 + -0.2986111111 + x = 1.582983027 + -0.2986111111 Combine like terms: 0.2986111111 + -0.2986111111 = 0.0000000000 0.0000000000 + x = 1.582983027 + -0.2986111111 x = 1.582983027 + -0.2986111111 Combine like terms: 1.582983027 + -0.2986111111 = 1.2843719159 x = 1.2843719159 Simplifying x = 1.2843719159Subproblem 2
x + 0.2986111111 = -1.582983027 Simplifying x + 0.2986111111 = -1.582983027 Reorder the terms: 0.2986111111 + x = -1.582983027 Solving 0.2986111111 + x = -1.582983027 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.2986111111' to each side of the equation. 0.2986111111 + -0.2986111111 + x = -1.582983027 + -0.2986111111 Combine like terms: 0.2986111111 + -0.2986111111 = 0.0000000000 0.0000000000 + x = -1.582983027 + -0.2986111111 x = -1.582983027 + -0.2986111111 Combine like terms: -1.582983027 + -0.2986111111 = -1.8815941381 x = -1.8815941381 Simplifying x = -1.8815941381Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.2843719159, -1.8815941381}
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