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Simplifying (15 + 13x) * x = 198 Reorder the terms for easier multiplication: x(15 + 13x) = 198 (15 * x + 13x * x) = 198 (15x + 13x2) = 198 Solving 15x + 13x2 = 198 Solving for variable 'x'. Reorder the terms: -198 + 15x + 13x2 = 198 + -198 Combine like terms: 198 + -198 = 0 -198 + 15x + 13x2 = 0 Begin completing the square. Divide all terms by 13 the coefficient of the squared term: Divide each side by '13'. -15.23076923 + 1.153846154x + x2 = 0 Move the constant term to the right: Add '15.23076923' to each side of the equation. -15.23076923 + 1.153846154x + 15.23076923 + x2 = 0 + 15.23076923 Reorder the terms: -15.23076923 + 15.23076923 + 1.153846154x + x2 = 0 + 15.23076923 Combine like terms: -15.23076923 + 15.23076923 = 0.00000000 0.00000000 + 1.153846154x + x2 = 0 + 15.23076923 1.153846154x + x2 = 0 + 15.23076923 Combine like terms: 0 + 15.23076923 = 15.23076923 1.153846154x + x2 = 15.23076923 The x term is 1.153846154x. Take half its coefficient (0.576923077). Square it (0.3328402368) and add it to both sides. Add '0.3328402368' to each side of the equation. 1.153846154x + 0.3328402368 + x2 = 15.23076923 + 0.3328402368 Reorder the terms: 0.3328402368 + 1.153846154x + x2 = 15.23076923 + 0.3328402368 Combine like terms: 15.23076923 + 0.3328402368 = 15.5636094668 0.3328402368 + 1.153846154x + x2 = 15.5636094668 Factor a perfect square on the left side: (x + 0.576923077)(x + 0.576923077) = 15.5636094668 Calculate the square root of the right side: 3.945074076 Break this problem into two subproblems by setting (x + 0.576923077) equal to 3.945074076 and -3.945074076.Subproblem 1
x + 0.576923077 = 3.945074076 Simplifying x + 0.576923077 = 3.945074076 Reorder the terms: 0.576923077 + x = 3.945074076 Solving 0.576923077 + x = 3.945074076 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.576923077' to each side of the equation. 0.576923077 + -0.576923077 + x = 3.945074076 + -0.576923077 Combine like terms: 0.576923077 + -0.576923077 = 0.000000000 0.000000000 + x = 3.945074076 + -0.576923077 x = 3.945074076 + -0.576923077 Combine like terms: 3.945074076 + -0.576923077 = 3.368150999 x = 3.368150999 Simplifying x = 3.368150999Subproblem 2
x + 0.576923077 = -3.945074076 Simplifying x + 0.576923077 = -3.945074076 Reorder the terms: 0.576923077 + x = -3.945074076 Solving 0.576923077 + x = -3.945074076 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.576923077' to each side of the equation. 0.576923077 + -0.576923077 + x = -3.945074076 + -0.576923077 Combine like terms: 0.576923077 + -0.576923077 = 0.000000000 0.000000000 + x = -3.945074076 + -0.576923077 x = -3.945074076 + -0.576923077 Combine like terms: -3.945074076 + -0.576923077 = -4.521997153 x = -4.521997153 Simplifying x = -4.521997153Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.368150999, -4.521997153}
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