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Simplifying 16a(2a + 3) = 8a + 20 Reorder the terms: 16a(3 + 2a) = 8a + 20 (3 * 16a + 2a * 16a) = 8a + 20 (48a + 32a2) = 8a + 20 Reorder the terms: 48a + 32a2 = 20 + 8a Solving 48a + 32a2 = 20 + 8a Solving for variable 'a'. Reorder the terms: -20 + 48a + -8a + 32a2 = 20 + 8a + -20 + -8a Combine like terms: 48a + -8a = 40a -20 + 40a + 32a2 = 20 + 8a + -20 + -8a Reorder the terms: -20 + 40a + 32a2 = 20 + -20 + 8a + -8a Combine like terms: 20 + -20 = 0 -20 + 40a + 32a2 = 0 + 8a + -8a -20 + 40a + 32a2 = 8a + -8a Combine like terms: 8a + -8a = 0 -20 + 40a + 32a2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-5 + 10a + 8a2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-5 + 10a + 8a2)' equal to zero and attempt to solve: Simplifying -5 + 10a + 8a2 = 0 Solving -5 + 10a + 8a2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.625 + 1.25a + a2 = 0 Move the constant term to the right: Add '0.625' to each side of the equation. -0.625 + 1.25a + 0.625 + a2 = 0 + 0.625 Reorder the terms: -0.625 + 0.625 + 1.25a + a2 = 0 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + 1.25a + a2 = 0 + 0.625 1.25a + a2 = 0 + 0.625 Combine like terms: 0 + 0.625 = 0.625 1.25a + a2 = 0.625 The a term is 1.25a. Take half its coefficient (0.625). Square it (0.390625) and add it to both sides. Add '0.390625' to each side of the equation. 1.25a + 0.390625 + a2 = 0.625 + 0.390625 Reorder the terms: 0.390625 + 1.25a + a2 = 0.625 + 0.390625 Combine like terms: 0.625 + 0.390625 = 1.015625 0.390625 + 1.25a + a2 = 1.015625 Factor a perfect square on the left side: (a + 0.625)(a + 0.625) = 1.015625 Calculate the square root of the right side: 1.007782219 Break this problem into two subproblems by setting (a + 0.625) equal to 1.007782219 and -1.007782219.Subproblem 1
a + 0.625 = 1.007782219 Simplifying a + 0.625 = 1.007782219 Reorder the terms: 0.625 + a = 1.007782219 Solving 0.625 + a = 1.007782219 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + a = 1.007782219 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + a = 1.007782219 + -0.625 a = 1.007782219 + -0.625 Combine like terms: 1.007782219 + -0.625 = 0.382782219 a = 0.382782219 Simplifying a = 0.382782219Subproblem 2
a + 0.625 = -1.007782219 Simplifying a + 0.625 = -1.007782219 Reorder the terms: 0.625 + a = -1.007782219 Solving 0.625 + a = -1.007782219 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + a = -1.007782219 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + a = -1.007782219 + -0.625 a = -1.007782219 + -0.625 Combine like terms: -1.007782219 + -0.625 = -1.632782219 a = -1.632782219 Simplifying a = -1.632782219Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.382782219, -1.632782219}Solution
a = {0.382782219, -1.632782219}
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