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Simplifying 2(3a + 2a * 6a + -4a) = 7a + 5 Reorder the terms for easier multiplication: 2(3a + 2 * 6a * a + -4a) = 7a + 5 Multiply 2 * 6 2(3a + 12a * a + -4a) = 7a + 5 Multiply a * a 2(3a + 12a2 + -4a) = 7a + 5 Reorder the terms: 2(3a + -4a + 12a2) = 7a + 5 Combine like terms: 3a + -4a = -1a 2(-1a + 12a2) = 7a + 5 (-1a * 2 + 12a2 * 2) = 7a + 5 (-2a + 24a2) = 7a + 5 Reorder the terms: -2a + 24a2 = 5 + 7a Solving -2a + 24a2 = 5 + 7a Solving for variable 'a'. Reorder the terms: -5 + -2a + -7a + 24a2 = 5 + 7a + -5 + -7a Combine like terms: -2a + -7a = -9a -5 + -9a + 24a2 = 5 + 7a + -5 + -7a Reorder the terms: -5 + -9a + 24a2 = 5 + -5 + 7a + -7a Combine like terms: 5 + -5 = 0 -5 + -9a + 24a2 = 0 + 7a + -7a -5 + -9a + 24a2 = 7a + -7a Combine like terms: 7a + -7a = 0 -5 + -9a + 24a2 = 0 Begin completing the square. Divide all terms by 24 the coefficient of the squared term: Divide each side by '24'. -0.2083333333 + -0.375a + a2 = 0 Move the constant term to the right: Add '0.2083333333' to each side of the equation. -0.2083333333 + -0.375a + 0.2083333333 + a2 = 0 + 0.2083333333 Reorder the terms: -0.2083333333 + 0.2083333333 + -0.375a + a2 = 0 + 0.2083333333 Combine like terms: -0.2083333333 + 0.2083333333 = 0.0000000000 0.0000000000 + -0.375a + a2 = 0 + 0.2083333333 -0.375a + a2 = 0 + 0.2083333333 Combine like terms: 0 + 0.2083333333 = 0.2083333333 -0.375a + a2 = 0.2083333333 The a term is -0.375a. Take half its coefficient (-0.1875). Square it (0.03515625) and add it to both sides. Add '0.03515625' to each side of the equation. -0.375a + 0.03515625 + a2 = 0.2083333333 + 0.03515625 Reorder the terms: 0.03515625 + -0.375a + a2 = 0.2083333333 + 0.03515625 Combine like terms: 0.2083333333 + 0.03515625 = 0.2434895833 0.03515625 + -0.375a + a2 = 0.2434895833 Factor a perfect square on the left side: (a + -0.1875)(a + -0.1875) = 0.2434895833 Calculate the square root of the right side: 0.493446637 Break this problem into two subproblems by setting (a + -0.1875) equal to 0.493446637 and -0.493446637.Subproblem 1
a + -0.1875 = 0.493446637 Simplifying a + -0.1875 = 0.493446637 Reorder the terms: -0.1875 + a = 0.493446637 Solving -0.1875 + a = 0.493446637 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.1875' to each side of the equation. -0.1875 + 0.1875 + a = 0.493446637 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + a = 0.493446637 + 0.1875 a = 0.493446637 + 0.1875 Combine like terms: 0.493446637 + 0.1875 = 0.680946637 a = 0.680946637 Simplifying a = 0.680946637Subproblem 2
a + -0.1875 = -0.493446637 Simplifying a + -0.1875 = -0.493446637 Reorder the terms: -0.1875 + a = -0.493446637 Solving -0.1875 + a = -0.493446637 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.1875' to each side of the equation. -0.1875 + 0.1875 + a = -0.493446637 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + a = -0.493446637 + 0.1875 a = -0.493446637 + 0.1875 Combine like terms: -0.493446637 + 0.1875 = -0.305946637 a = -0.305946637 Simplifying a = -0.305946637Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.680946637, -0.305946637}
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