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Simplifying 2(a + -4) = 4a(2a + -4) Reorder the terms: 2(-4 + a) = 4a(2a + -4) (-4 * 2 + a * 2) = 4a(2a + -4) (-8 + 2a) = 4a(2a + -4) Reorder the terms: -8 + 2a = 4a(-4 + 2a) -8 + 2a = (-4 * 4a + 2a * 4a) -8 + 2a = (-16a + 8a2) Solving -8 + 2a = -16a + 8a2 Solving for variable 'a'. Combine like terms: 2a + 16a = 18a -8 + 18a + -8a2 = -16a + 8a2 + 16a + -8a2 Reorder the terms: -8 + 18a + -8a2 = -16a + 16a + 8a2 + -8a2 Combine like terms: -16a + 16a = 0 -8 + 18a + -8a2 = 0 + 8a2 + -8a2 -8 + 18a + -8a2 = 8a2 + -8a2 Combine like terms: 8a2 + -8a2 = 0 -8 + 18a + -8a2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-4 + 9a + -4a2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-4 + 9a + -4a2)' equal to zero and attempt to solve: Simplifying -4 + 9a + -4a2 = 0 Solving -4 + 9a + -4a2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. 1 + -2.25a + a2 = 0 Move the constant term to the right: Add '-1' to each side of the equation. 1 + -2.25a + -1 + a2 = 0 + -1 Reorder the terms: 1 + -1 + -2.25a + a2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -2.25a + a2 = 0 + -1 -2.25a + a2 = 0 + -1 Combine like terms: 0 + -1 = -1 -2.25a + a2 = -1 The a term is -2.25a. Take half its coefficient (-1.125). Square it (1.265625) and add it to both sides. Add '1.265625' to each side of the equation. -2.25a + 1.265625 + a2 = -1 + 1.265625 Reorder the terms: 1.265625 + -2.25a + a2 = -1 + 1.265625 Combine like terms: -1 + 1.265625 = 0.265625 1.265625 + -2.25a + a2 = 0.265625 Factor a perfect square on the left side: (a + -1.125)(a + -1.125) = 0.265625 Calculate the square root of the right side: 0.515388203 Break this problem into two subproblems by setting (a + -1.125) equal to 0.515388203 and -0.515388203.Subproblem 1
a + -1.125 = 0.515388203 Simplifying a + -1.125 = 0.515388203 Reorder the terms: -1.125 + a = 0.515388203 Solving -1.125 + a = 0.515388203 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.125' to each side of the equation. -1.125 + 1.125 + a = 0.515388203 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + a = 0.515388203 + 1.125 a = 0.515388203 + 1.125 Combine like terms: 0.515388203 + 1.125 = 1.640388203 a = 1.640388203 Simplifying a = 1.640388203Subproblem 2
a + -1.125 = -0.515388203 Simplifying a + -1.125 = -0.515388203 Reorder the terms: -1.125 + a = -0.515388203 Solving -1.125 + a = -0.515388203 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.125' to each side of the equation. -1.125 + 1.125 + a = -0.515388203 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + a = -0.515388203 + 1.125 a = -0.515388203 + 1.125 Combine like terms: -0.515388203 + 1.125 = 0.609611797 a = 0.609611797 Simplifying a = 0.609611797Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.640388203, 0.609611797}Solution
a = {1.640388203, 0.609611797}
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