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Simplifying 5x(3 + 6x) = 7 + 12 (3 * 5x + 6x * 5x) = 7 + 12 (15x + 30x2) = 7 + 12 Combine like terms: 7 + 12 = 19 15x + 30x2 = 19 Solving 15x + 30x2 = 19 Solving for variable 'x'. Reorder the terms: -19 + 15x + 30x2 = 19 + -19 Combine like terms: 19 + -19 = 0 -19 + 15x + 30x2 = 0 Begin completing the square. Divide all terms by 30 the coefficient of the squared term: Divide each side by '30'. -0.6333333333 + 0.5x + x2 = 0.0 Move the constant term to the right: Add '0.6333333333' to each side of the equation. -0.6333333333 + 0.5x + 0.6333333333 + x2 = 0.0 + 0.6333333333 Reorder the terms: -0.6333333333 + 0.6333333333 + 0.5x + x2 = 0.0 + 0.6333333333 Combine like terms: -0.6333333333 + 0.6333333333 = 0.0000000000 0.0000000000 + 0.5x + x2 = 0.0 + 0.6333333333 0.5x + x2 = 0.0 + 0.6333333333 Combine like terms: 0.0 + 0.6333333333 = 0.6333333333 0.5x + x2 = 0.6333333333 The x term is 0.5x. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5x + 0.0625 + x2 = 0.6333333333 + 0.0625 Reorder the terms: 0.0625 + 0.5x + x2 = 0.6333333333 + 0.0625 Combine like terms: 0.6333333333 + 0.0625 = 0.6958333333 0.0625 + 0.5x + x2 = 0.6958333333 Factor a perfect square on the left side: (x + 0.25)(x + 0.25) = 0.6958333333 Calculate the square root of the right side: 0.83416625 Break this problem into two subproblems by setting (x + 0.25) equal to 0.83416625 and -0.83416625.Subproblem 1
x + 0.25 = 0.83416625 Simplifying x + 0.25 = 0.83416625 Reorder the terms: 0.25 + x = 0.83416625 Solving 0.25 + x = 0.83416625 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = 0.83416625 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 0.83416625 + -0.25 x = 0.83416625 + -0.25 Combine like terms: 0.83416625 + -0.25 = 0.58416625 x = 0.58416625 Simplifying x = 0.58416625Subproblem 2
x + 0.25 = -0.83416625 Simplifying x + 0.25 = -0.83416625 Reorder the terms: 0.25 + x = -0.83416625 Solving 0.25 + x = -0.83416625 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = -0.83416625 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -0.83416625 + -0.25 x = -0.83416625 + -0.25 Combine like terms: -0.83416625 + -0.25 = -1.08416625 x = -1.08416625 Simplifying x = -1.08416625Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.58416625, -1.08416625}
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