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Simplifying 6(5x + 6) * 3x + 3 = 103 Reorder the terms: 6(6 + 5x) * 3x + 3 = 103 Reorder the terms for easier multiplication: 6 * 3x(6 + 5x) + 3 = 103 Multiply 6 * 3 18x(6 + 5x) + 3 = 103 (6 * 18x + 5x * 18x) + 3 = 103 (108x + 90x2) + 3 = 103 Reorder the terms: 3 + 108x + 90x2 = 103 Solving 3 + 108x + 90x2 = 103 Solving for variable 'x'. Reorder the terms: 3 + -103 + 108x + 90x2 = 103 + -103 Combine like terms: 3 + -103 = -100 -100 + 108x + 90x2 = 103 + -103 Combine like terms: 103 + -103 = 0 -100 + 108x + 90x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-50 + 54x + 45x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-50 + 54x + 45x2)' equal to zero and attempt to solve: Simplifying -50 + 54x + 45x2 = 0 Solving -50 + 54x + 45x2 = 0 Begin completing the square. Divide all terms by 45 the coefficient of the squared term: Divide each side by '45'. -1.111111111 + 1.2x + x2 = 0 Move the constant term to the right: Add '1.111111111' to each side of the equation. -1.111111111 + 1.2x + 1.111111111 + x2 = 0 + 1.111111111 Reorder the terms: -1.111111111 + 1.111111111 + 1.2x + x2 = 0 + 1.111111111 Combine like terms: -1.111111111 + 1.111111111 = 0.000000000 0.000000000 + 1.2x + x2 = 0 + 1.111111111 1.2x + x2 = 0 + 1.111111111 Combine like terms: 0 + 1.111111111 = 1.111111111 1.2x + x2 = 1.111111111 The x term is 1.2x. Take half its coefficient (0.6). Square it (0.36) and add it to both sides. Add '0.36' to each side of the equation. 1.2x + 0.36 + x2 = 1.111111111 + 0.36 Reorder the terms: 0.36 + 1.2x + x2 = 1.111111111 + 0.36 Combine like terms: 1.111111111 + 0.36 = 1.471111111 0.36 + 1.2x + x2 = 1.471111111 Factor a perfect square on the left side: (x + 0.6)(x + 0.6) = 1.471111111 Calculate the square root of the right side: 1.212893693 Break this problem into two subproblems by setting (x + 0.6) equal to 1.212893693 and -1.212893693.Subproblem 1
x + 0.6 = 1.212893693 Simplifying x + 0.6 = 1.212893693 Reorder the terms: 0.6 + x = 1.212893693 Solving 0.6 + x = 1.212893693 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.6' to each side of the equation. 0.6 + -0.6 + x = 1.212893693 + -0.6 Combine like terms: 0.6 + -0.6 = 0.0 0.0 + x = 1.212893693 + -0.6 x = 1.212893693 + -0.6 Combine like terms: 1.212893693 + -0.6 = 0.612893693 x = 0.612893693 Simplifying x = 0.612893693Subproblem 2
x + 0.6 = -1.212893693 Simplifying x + 0.6 = -1.212893693 Reorder the terms: 0.6 + x = -1.212893693 Solving 0.6 + x = -1.212893693 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.6' to each side of the equation. 0.6 + -0.6 + x = -1.212893693 + -0.6 Combine like terms: 0.6 + -0.6 = 0.0 0.0 + x = -1.212893693 + -0.6 x = -1.212893693 + -0.6 Combine like terms: -1.212893693 + -0.6 = -1.812893693 x = -1.812893693 Simplifying x = -1.812893693Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.612893693, -1.812893693}Solution
x = {0.612893693, -1.812893693}
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