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Simplifying 8a2 + 16a + -1 = 7 Reorder the terms: -1 + 16a + 8a2 = 7 Solving -1 + 16a + 8a2 = 7 Solving for variable 'a'. Reorder the terms: -1 + -7 + 16a + 8a2 = 7 + -7 Combine like terms: -1 + -7 = -8 -8 + 16a + 8a2 = 7 + -7 Combine like terms: 7 + -7 = 0 -8 + 16a + 8a2 = 0 Factor out the Greatest Common Factor (GCF), '8'. 8(-1 + 2a + a2) = 0 Ignore the factor 8.Subproblem 1
Set the factor '(-1 + 2a + a2)' equal to zero and attempt to solve: Simplifying -1 + 2a + a2 = 0 Solving -1 + 2a + a2 = 0 Begin completing the square. Move the constant term to the right: Add '1' to each side of the equation. -1 + 2a + 1 + a2 = 0 + 1 Reorder the terms: -1 + 1 + 2a + a2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 2a + a2 = 0 + 1 2a + a2 = 0 + 1 Combine like terms: 0 + 1 = 1 2a + a2 = 1 The a term is 2a. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2a + 1 + a2 = 1 + 1 Reorder the terms: 1 + 2a + a2 = 1 + 1 Combine like terms: 1 + 1 = 2 1 + 2a + a2 = 2 Factor a perfect square on the left side: (a + 1)(a + 1) = 2 Calculate the square root of the right side: 1.414213562 Break this problem into two subproblems by setting (a + 1) equal to 1.414213562 and -1.414213562.Subproblem 1
a + 1 = 1.414213562 Simplifying a + 1 = 1.414213562 Reorder the terms: 1 + a = 1.414213562 Solving 1 + a = 1.414213562 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + a = 1.414213562 + -1 Combine like terms: 1 + -1 = 0 0 + a = 1.414213562 + -1 a = 1.414213562 + -1 Combine like terms: 1.414213562 + -1 = 0.414213562 a = 0.414213562 Simplifying a = 0.414213562Subproblem 2
a + 1 = -1.414213562 Simplifying a + 1 = -1.414213562 Reorder the terms: 1 + a = -1.414213562 Solving 1 + a = -1.414213562 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + a = -1.414213562 + -1 Combine like terms: 1 + -1 = 0 0 + a = -1.414213562 + -1 a = -1.414213562 + -1 Combine like terms: -1.414213562 + -1 = -2.414213562 a = -2.414213562 Simplifying a = -2.414213562Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.414213562, -2.414213562}Solution
a = {0.414213562, -2.414213562}
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