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Simplifying 3a(a + 4) + -2 + 7 = 0 Reorder the terms: 3a(4 + a) + -2 + 7 = 0 (4 * 3a + a * 3a) + -2 + 7 = 0 (12a + 3a2) + -2 + 7 = 0 Reorder the terms: -2 + 7 + 12a + 3a2 = 0 Combine like terms: -2 + 7 = 5 5 + 12a + 3a2 = 0 Solving 5 + 12a + 3a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 1.666666667 + 4a + a2 = 0 Move the constant term to the right: Add '-1.666666667' to each side of the equation. 1.666666667 + 4a + -1.666666667 + a2 = 0 + -1.666666667 Reorder the terms: 1.666666667 + -1.666666667 + 4a + a2 = 0 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + 4a + a2 = 0 + -1.666666667 4a + a2 = 0 + -1.666666667 Combine like terms: 0 + -1.666666667 = -1.666666667 4a + a2 = -1.666666667 The a term is 4a. Take half its coefficient (2). Square it (4) and add it to both sides. Add '4' to each side of the equation. 4a + 4 + a2 = -1.666666667 + 4 Reorder the terms: 4 + 4a + a2 = -1.666666667 + 4 Combine like terms: -1.666666667 + 4 = 2.333333333 4 + 4a + a2 = 2.333333333 Factor a perfect square on the left side: (a + 2)(a + 2) = 2.333333333 Calculate the square root of the right side: 1.527525232 Break this problem into two subproblems by setting (a + 2) equal to 1.527525232 and -1.527525232.Subproblem 1
a + 2 = 1.527525232 Simplifying a + 2 = 1.527525232 Reorder the terms: 2 + a = 1.527525232 Solving 2 + a = 1.527525232 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + a = 1.527525232 + -2 Combine like terms: 2 + -2 = 0 0 + a = 1.527525232 + -2 a = 1.527525232 + -2 Combine like terms: 1.527525232 + -2 = -0.472474768 a = -0.472474768 Simplifying a = -0.472474768Subproblem 2
a + 2 = -1.527525232 Simplifying a + 2 = -1.527525232 Reorder the terms: 2 + a = -1.527525232 Solving 2 + a = -1.527525232 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + a = -1.527525232 + -2 Combine like terms: 2 + -2 = 0 0 + a = -1.527525232 + -2 a = -1.527525232 + -2 Combine like terms: -1.527525232 + -2 = -3.527525232 a = -3.527525232 Simplifying a = -3.527525232Solution
The solution to the problem is based on the solutions from the subproblems. a = {-0.472474768, -3.527525232}
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