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Simplifying 3a(a + 4) * 12 + -2a + 7 = 0 Reorder the terms: 3a(4 + a) * 12 + -2a + 7 = 0 Reorder the terms for easier multiplication: 3 * 12a(4 + a) + -2a + 7 = 0 Multiply 3 * 12 36a(4 + a) + -2a + 7 = 0 (4 * 36a + a * 36a) + -2a + 7 = 0 (144a + 36a2) + -2a + 7 = 0 Reorder the terms: 7 + 144a + -2a + 36a2 = 0 Combine like terms: 144a + -2a = 142a 7 + 142a + 36a2 = 0 Solving 7 + 142a + 36a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 36 the coefficient of the squared term: Divide each side by '36'. 0.1944444444 + 3.944444444a + a2 = 0 Move the constant term to the right: Add '-0.1944444444' to each side of the equation. 0.1944444444 + 3.944444444a + -0.1944444444 + a2 = 0 + -0.1944444444 Reorder the terms: 0.1944444444 + -0.1944444444 + 3.944444444a + a2 = 0 + -0.1944444444 Combine like terms: 0.1944444444 + -0.1944444444 = 0.0000000000 0.0000000000 + 3.944444444a + a2 = 0 + -0.1944444444 3.944444444a + a2 = 0 + -0.1944444444 Combine like terms: 0 + -0.1944444444 = -0.1944444444 3.944444444a + a2 = -0.1944444444 The a term is 3.944444444a. Take half its coefficient (1.972222222). Square it (3.889660493) and add it to both sides. Add '3.889660493' to each side of the equation. 3.944444444a + 3.889660493 + a2 = -0.1944444444 + 3.889660493 Reorder the terms: 3.889660493 + 3.944444444a + a2 = -0.1944444444 + 3.889660493 Combine like terms: -0.1944444444 + 3.889660493 = 3.6952160486 3.889660493 + 3.944444444a + a2 = 3.6952160486 Factor a perfect square on the left side: (a + 1.972222222)(a + 1.972222222) = 3.6952160486 Calculate the square root of the right side: 1.922294475 Break this problem into two subproblems by setting (a + 1.972222222) equal to 1.922294475 and -1.922294475.Subproblem 1
a + 1.972222222 = 1.922294475 Simplifying a + 1.972222222 = 1.922294475 Reorder the terms: 1.972222222 + a = 1.922294475 Solving 1.972222222 + a = 1.922294475 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.972222222' to each side of the equation. 1.972222222 + -1.972222222 + a = 1.922294475 + -1.972222222 Combine like terms: 1.972222222 + -1.972222222 = 0.000000000 0.000000000 + a = 1.922294475 + -1.972222222 a = 1.922294475 + -1.972222222 Combine like terms: 1.922294475 + -1.972222222 = -0.049927747 a = -0.049927747 Simplifying a = -0.049927747Subproblem 2
a + 1.972222222 = -1.922294475 Simplifying a + 1.972222222 = -1.922294475 Reorder the terms: 1.972222222 + a = -1.922294475 Solving 1.972222222 + a = -1.922294475 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.972222222' to each side of the equation. 1.972222222 + -1.972222222 + a = -1.922294475 + -1.972222222 Combine like terms: 1.972222222 + -1.972222222 = 0.000000000 0.000000000 + a = -1.922294475 + -1.972222222 a = -1.922294475 + -1.972222222 Combine like terms: -1.922294475 + -1.972222222 = -3.894516697 a = -3.894516697 Simplifying a = -3.894516697Solution
The solution to the problem is based on the solutions from the subproblems. a = {-0.049927747, -3.894516697}
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