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Simplifying 3a2 + -12 = -7a Reorder the terms: -12 + 3a2 = -7a Solving -12 + 3a2 = -7a Solving for variable 'a'. Reorder the terms: -12 + 7a + 3a2 = -7a + 7a Combine like terms: -7a + 7a = 0 -12 + 7a + 3a2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -4 + 2.333333333a + a2 = 0 Move the constant term to the right: Add '4' to each side of the equation. -4 + 2.333333333a + 4 + a2 = 0 + 4 Reorder the terms: -4 + 4 + 2.333333333a + a2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + 2.333333333a + a2 = 0 + 4 2.333333333a + a2 = 0 + 4 Combine like terms: 0 + 4 = 4 2.333333333a + a2 = 4 The a term is 2.333333333a. Take half its coefficient (1.166666667). Square it (1.361111112) and add it to both sides. Add '1.361111112' to each side of the equation. 2.333333333a + 1.361111112 + a2 = 4 + 1.361111112 Reorder the terms: 1.361111112 + 2.333333333a + a2 = 4 + 1.361111112 Combine like terms: 4 + 1.361111112 = 5.361111112 1.361111112 + 2.333333333a + a2 = 5.361111112 Factor a perfect square on the left side: (a + 1.166666667)(a + 1.166666667) = 5.361111112 Calculate the square root of the right side: 2.315407332 Break this problem into two subproblems by setting (a + 1.166666667) equal to 2.315407332 and -2.315407332.Subproblem 1
a + 1.166666667 = 2.315407332 Simplifying a + 1.166666667 = 2.315407332 Reorder the terms: 1.166666667 + a = 2.315407332 Solving 1.166666667 + a = 2.315407332 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.166666667' to each side of the equation. 1.166666667 + -1.166666667 + a = 2.315407332 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + a = 2.315407332 + -1.166666667 a = 2.315407332 + -1.166666667 Combine like terms: 2.315407332 + -1.166666667 = 1.148740665 a = 1.148740665 Simplifying a = 1.148740665Subproblem 2
a + 1.166666667 = -2.315407332 Simplifying a + 1.166666667 = -2.315407332 Reorder the terms: 1.166666667 + a = -2.315407332 Solving 1.166666667 + a = -2.315407332 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.166666667' to each side of the equation. 1.166666667 + -1.166666667 + a = -2.315407332 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + a = -2.315407332 + -1.166666667 a = -2.315407332 + -1.166666667 Combine like terms: -2.315407332 + -1.166666667 = -3.482073999 a = -3.482073999 Simplifying a = -3.482073999Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.148740665, -3.482073999}
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