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Simplifying 125(a) = 0.01a2 + 0.05a + 107 Reorder the terms: 125a = 107 + 0.05a + 0.01a2 Solving 125a = 107 + 0.05a + 0.01a2 Solving for variable 'a'. Reorder the terms: -107 + 125a + -0.05a + -0.01a2 = 107 + 0.05a + 0.01a2 + -107 + -0.05a + -0.01a2 Combine like terms: 125a + -0.05a = 124.95a -107 + 124.95a + -0.01a2 = 107 + 0.05a + 0.01a2 + -107 + -0.05a + -0.01a2 Reorder the terms: -107 + 124.95a + -0.01a2 = 107 + -107 + 0.05a + -0.05a + 0.01a2 + -0.01a2 Combine like terms: 107 + -107 = 0 -107 + 124.95a + -0.01a2 = 0 + 0.05a + -0.05a + 0.01a2 + -0.01a2 -107 + 124.95a + -0.01a2 = 0.05a + -0.05a + 0.01a2 + -0.01a2 Combine like terms: 0.05a + -0.05a = 0.00 -107 + 124.95a + -0.01a2 = 0.00 + 0.01a2 + -0.01a2 -107 + 124.95a + -0.01a2 = 0.01a2 + -0.01a2 Combine like terms: 0.01a2 + -0.01a2 = 0.00 -107 + 124.95a + -0.01a2 = 0.00 Begin completing the square. Divide all terms by -0.01 the coefficient of the squared term: Divide each side by '-0.01'. 10700 + -12495a + a2 = 0 Move the constant term to the right: Add '-10700' to each side of the equation. 10700 + -12495a + -10700 + a2 = 0 + -10700 Reorder the terms: 10700 + -10700 + -12495a + a2 = 0 + -10700 Combine like terms: 10700 + -10700 = 0 0 + -12495a + a2 = 0 + -10700 -12495a + a2 = 0 + -10700 Combine like terms: 0 + -10700 = -10700 -12495a + a2 = -10700 The a term is -12495a. Take half its coefficient (-6247.5). Square it (39031256.25) and add it to both sides. Add '39031256.25' to each side of the equation. -12495a + 39031256.25 + a2 = -10700 + 39031256.25 Reorder the terms: 39031256.25 + -12495a + a2 = -10700 + 39031256.25 Combine like terms: -10700 + 39031256.25 = 39020556.25 39031256.25 + -12495a + a2 = 39020556.25 Factor a perfect square on the left side: (a + -6247.5)(a + -6247.5) = 39020556.25 Calculate the square root of the right side: 6246.643598766 Break this problem into two subproblems by setting (a + -6247.5) equal to 6246.643598766 and -6246.643598766.Subproblem 1
a + -6247.5 = 6246.643598766 Simplifying a + -6247.5 = 6246.643598766 Reorder the terms: -6247.5 + a = 6246.643598766 Solving -6247.5 + a = 6246.643598766 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '6247.5' to each side of the equation. -6247.5 + 6247.5 + a = 6246.643598766 + 6247.5 Combine like terms: -6247.5 + 6247.5 = 0.0 0.0 + a = 6246.643598766 + 6247.5 a = 6246.643598766 + 6247.5 Combine like terms: 6246.643598766 + 6247.5 = 12494.143598766 a = 12494.143598766 Simplifying a = 12494.143598766Subproblem 2
a + -6247.5 = -6246.643598766 Simplifying a + -6247.5 = -6246.643598766 Reorder the terms: -6247.5 + a = -6246.643598766 Solving -6247.5 + a = -6246.643598766 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '6247.5' to each side of the equation. -6247.5 + 6247.5 + a = -6246.643598766 + 6247.5 Combine like terms: -6247.5 + 6247.5 = 0.0 0.0 + a = -6246.643598766 + 6247.5 a = -6246.643598766 + 6247.5 Combine like terms: -6246.643598766 + 6247.5 = 0.856401234 a = 0.856401234 Simplifying a = 0.856401234Solution
The solution to the problem is based on the solutions from the subproblems. a = {12494.143598766, 0.856401234}
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