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Simplifying a2 + -0.05a + 0.00006 = 1 Reorder the terms: 0.00006 + -0.05a + a2 = 1 Solving 0.00006 + -0.05a + a2 = 1 Solving for variable 'a'. Reorder the terms: 0.00006 + -1 + -0.05a + a2 = 1 + -1 Combine like terms: 0.00006 + -1 = -0.99994 -0.99994 + -0.05a + a2 = 1 + -1 Combine like terms: 1 + -1 = 0 -0.99994 + -0.05a + a2 = 0 Begin completing the square. Move the constant term to the right: Add '0.99994' to each side of the equation. -0.99994 + -0.05a + 0.99994 + a2 = 0 + 0.99994 Reorder the terms: -0.99994 + 0.99994 + -0.05a + a2 = 0 + 0.99994 Combine like terms: -0.99994 + 0.99994 = 0.00000 0.00000 + -0.05a + a2 = 0 + 0.99994 -0.05a + a2 = 0 + 0.99994 Combine like terms: 0 + 0.99994 = 0.99994 -0.05a + a2 = 0.99994 The a term is -0.05a. Take half its coefficient (-0.025). Square it (0.000625) and add it to both sides. Add '0.000625' to each side of the equation. -0.05a + 0.000625 + a2 = 0.99994 + 0.000625 Reorder the terms: 0.000625 + -0.05a + a2 = 0.99994 + 0.000625 Combine like terms: 0.99994 + 0.000625 = 1.000565 0.000625 + -0.05a + a2 = 1.000565 Factor a perfect square on the left side: (a + -0.025)(a + -0.025) = 1.000565 Calculate the square root of the right side: 1.00028246 Break this problem into two subproblems by setting (a + -0.025) equal to 1.00028246 and -1.00028246.Subproblem 1
a + -0.025 = 1.00028246 Simplifying a + -0.025 = 1.00028246 Reorder the terms: -0.025 + a = 1.00028246 Solving -0.025 + a = 1.00028246 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.025' to each side of the equation. -0.025 + 0.025 + a = 1.00028246 + 0.025 Combine like terms: -0.025 + 0.025 = 0.000 0.000 + a = 1.00028246 + 0.025 a = 1.00028246 + 0.025 Combine like terms: 1.00028246 + 0.025 = 1.02528246 a = 1.02528246 Simplifying a = 1.02528246Subproblem 2
a + -0.025 = -1.00028246 Simplifying a + -0.025 = -1.00028246 Reorder the terms: -0.025 + a = -1.00028246 Solving -0.025 + a = -1.00028246 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.025' to each side of the equation. -0.025 + 0.025 + a = -1.00028246 + 0.025 Combine like terms: -0.025 + 0.025 = 0.000 0.000 + a = -1.00028246 + 0.025 a = -1.00028246 + 0.025 Combine like terms: -1.00028246 + 0.025 = -0.97528246 a = -0.97528246 Simplifying a = -0.97528246Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.02528246, -0.97528246}
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