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Simplifying 104 = 4a(3a + -1) Reorder the terms: 104 = 4a(-1 + 3a) 104 = (-1 * 4a + 3a * 4a) 104 = (-4a + 12a2) Solving 104 = -4a + 12a2 Solving for variable 'a'. Reorder the terms: 104 + 4a + -12a2 = -4a + 4a + 12a2 + -12a2 Combine like terms: -4a + 4a = 0 104 + 4a + -12a2 = 0 + 12a2 + -12a2 104 + 4a + -12a2 = 12a2 + -12a2 Combine like terms: 12a2 + -12a2 = 0 104 + 4a + -12a2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(26 + a + -3a2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(26 + a + -3a2)' equal to zero and attempt to solve: Simplifying 26 + a + -3a2 = 0 Solving 26 + a + -3a2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -8.666666667 + -0.3333333333a + a2 = 0 Move the constant term to the right: Add '8.666666667' to each side of the equation. -8.666666667 + -0.3333333333a + 8.666666667 + a2 = 0 + 8.666666667 Reorder the terms: -8.666666667 + 8.666666667 + -0.3333333333a + a2 = 0 + 8.666666667 Combine like terms: -8.666666667 + 8.666666667 = 0.000000000 0.000000000 + -0.3333333333a + a2 = 0 + 8.666666667 -0.3333333333a + a2 = 0 + 8.666666667 Combine like terms: 0 + 8.666666667 = 8.666666667 -0.3333333333a + a2 = 8.666666667 The a term is a. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -0.3333333333a + 0.25 + a2 = 8.666666667 + 0.25 Reorder the terms: 0.25 + -0.3333333333a + a2 = 8.666666667 + 0.25 Combine like terms: 8.666666667 + 0.25 = 8.916666667 0.25 + -0.3333333333a + a2 = 8.916666667 Factor a perfect square on the left side: (a + 0.5)(a + 0.5) = 8.916666667 Calculate the square root of the right side: 2.986078811 Break this problem into two subproblems by setting (a + 0.5) equal to 2.986078811 and -2.986078811.Subproblem 1
a + 0.5 = 2.986078811 Simplifying a + 0.5 = 2.986078811 Reorder the terms: 0.5 + a = 2.986078811 Solving 0.5 + a = 2.986078811 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + a = 2.986078811 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = 2.986078811 + -0.5 a = 2.986078811 + -0.5 Combine like terms: 2.986078811 + -0.5 = 2.486078811 a = 2.486078811 Simplifying a = 2.486078811Subproblem 2
a + 0.5 = -2.986078811 Simplifying a + 0.5 = -2.986078811 Reorder the terms: 0.5 + a = -2.986078811 Solving 0.5 + a = -2.986078811 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + a = -2.986078811 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = -2.986078811 + -0.5 a = -2.986078811 + -0.5 Combine like terms: -2.986078811 + -0.5 = -3.486078811 a = -3.486078811 Simplifying a = -3.486078811Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.486078811, -3.486078811}Solution
a = {2.486078811, -3.486078811}
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