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Simplifying (2 + 3a)(1 + 4a) = 117 Multiply (2 + 3a) * (1 + 4a) (2(1 + 4a) + 3a * (1 + 4a)) = 117 ((1 * 2 + 4a * 2) + 3a * (1 + 4a)) = 117 ((2 + 8a) + 3a * (1 + 4a)) = 117 (2 + 8a + (1 * 3a + 4a * 3a)) = 117 (2 + 8a + (3a + 12a2)) = 117 Combine like terms: 8a + 3a = 11a (2 + 11a + 12a2) = 117 Solving 2 + 11a + 12a2 = 117 Solving for variable 'a'. Reorder the terms: 2 + -117 + 11a + 12a2 = 117 + -117 Combine like terms: 2 + -117 = -115 -115 + 11a + 12a2 = 117 + -117 Combine like terms: 117 + -117 = 0 -115 + 11a + 12a2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -9.583333333 + 0.9166666667a + a2 = 0 Move the constant term to the right: Add '9.583333333' to each side of the equation. -9.583333333 + 0.9166666667a + 9.583333333 + a2 = 0 + 9.583333333 Reorder the terms: -9.583333333 + 9.583333333 + 0.9166666667a + a2 = 0 + 9.583333333 Combine like terms: -9.583333333 + 9.583333333 = 0.000000000 0.000000000 + 0.9166666667a + a2 = 0 + 9.583333333 0.9166666667a + a2 = 0 + 9.583333333 Combine like terms: 0 + 9.583333333 = 9.583333333 0.9166666667a + a2 = 9.583333333 The a term is 0.9166666667a. Take half its coefficient (0.4583333334). Square it (0.2100694445) and add it to both sides. Add '0.2100694445' to each side of the equation. 0.9166666667a + 0.2100694445 + a2 = 9.583333333 + 0.2100694445 Reorder the terms: 0.2100694445 + 0.9166666667a + a2 = 9.583333333 + 0.2100694445 Combine like terms: 9.583333333 + 0.2100694445 = 9.7934027775 0.2100694445 + 0.9166666667a + a2 = 9.7934027775 Factor a perfect square on the left side: (a + 0.4583333334)(a + 0.4583333334) = 9.7934027775 Calculate the square root of the right side: 3.129441288 Break this problem into two subproblems by setting (a + 0.4583333334) equal to 3.129441288 and -3.129441288.Subproblem 1
a + 0.4583333334 = 3.129441288 Simplifying a + 0.4583333334 = 3.129441288 Reorder the terms: 0.4583333334 + a = 3.129441288 Solving 0.4583333334 + a = 3.129441288 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.4583333334' to each side of the equation. 0.4583333334 + -0.4583333334 + a = 3.129441288 + -0.4583333334 Combine like terms: 0.4583333334 + -0.4583333334 = 0.0000000000 0.0000000000 + a = 3.129441288 + -0.4583333334 a = 3.129441288 + -0.4583333334 Combine like terms: 3.129441288 + -0.4583333334 = 2.6711079546 a = 2.6711079546 Simplifying a = 2.6711079546Subproblem 2
a + 0.4583333334 = -3.129441288 Simplifying a + 0.4583333334 = -3.129441288 Reorder the terms: 0.4583333334 + a = -3.129441288 Solving 0.4583333334 + a = -3.129441288 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.4583333334' to each side of the equation. 0.4583333334 + -0.4583333334 + a = -3.129441288 + -0.4583333334 Combine like terms: 0.4583333334 + -0.4583333334 = 0.0000000000 0.0000000000 + a = -3.129441288 + -0.4583333334 a = -3.129441288 + -0.4583333334 Combine like terms: -3.129441288 + -0.4583333334 = -3.5877746214 a = -3.5877746214 Simplifying a = -3.5877746214Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.6711079546, -3.5877746214}
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