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Simplifying 34 = 0.0021a2 + -0.286a + 16.28 Reorder the terms: 34 = 16.28 + -0.286a + 0.0021a2 Solving 34 = 16.28 + -0.286a + 0.0021a2 Solving for variable 'a'. Combine like terms: 34 + -16.28 = 17.72 17.72 + 0.286a + -0.0021a2 = 16.28 + -0.286a + 0.0021a2 + -16.28 + 0.286a + -0.0021a2 Reorder the terms: 17.72 + 0.286a + -0.0021a2 = 16.28 + -16.28 + -0.286a + 0.286a + 0.0021a2 + -0.0021a2 Combine like terms: 16.28 + -16.28 = 0.00 17.72 + 0.286a + -0.0021a2 = 0.00 + -0.286a + 0.286a + 0.0021a2 + -0.0021a2 17.72 + 0.286a + -0.0021a2 = -0.286a + 0.286a + 0.0021a2 + -0.0021a2 Combine like terms: -0.286a + 0.286a = 0.000 17.72 + 0.286a + -0.0021a2 = 0.000 + 0.0021a2 + -0.0021a2 17.72 + 0.286a + -0.0021a2 = 0.0021a2 + -0.0021a2 Combine like terms: 0.0021a2 + -0.0021a2 = 0.0000 17.72 + 0.286a + -0.0021a2 = 0.0000 Begin completing the square. Divide all terms by -0.0021 the coefficient of the squared term: Divide each side by '-0.0021'. -8438.095238 + -136.1904762a + a2 = 0 Move the constant term to the right: Add '8438.095238' to each side of the equation. -8438.095238 + -136.1904762a + 8438.095238 + a2 = 0 + 8438.095238 Reorder the terms: -8438.095238 + 8438.095238 + -136.1904762a + a2 = 0 + 8438.095238 Combine like terms: -8438.095238 + 8438.095238 = 0.000000 0.000000 + -136.1904762a + a2 = 0 + 8438.095238 -136.1904762a + a2 = 0 + 8438.095238 Combine like terms: 0 + 8438.095238 = 8438.095238 -136.1904762a + a2 = 8438.095238 The a term is -136.1904762a. Take half its coefficient (-68.0952381). Square it (4636.961452) and add it to both sides. Add '4636.961452' to each side of the equation. -136.1904762a + 4636.961452 + a2 = 8438.095238 + 4636.961452 Reorder the terms: 4636.961452 + -136.1904762a + a2 = 8438.095238 + 4636.961452 Combine like terms: 8438.095238 + 4636.961452 = 13075.05669 4636.961452 + -136.1904762a + a2 = 13075.05669 Factor a perfect square on the left side: (a + -68.0952381)(a + -68.0952381) = 13075.05669 Calculate the square root of the right side: 114.346214148 Break this problem into two subproblems by setting (a + -68.0952381) equal to 114.346214148 and -114.346214148.Subproblem 1
a + -68.0952381 = 114.346214148 Simplifying a + -68.0952381 = 114.346214148 Reorder the terms: -68.0952381 + a = 114.346214148 Solving -68.0952381 + a = 114.346214148 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '68.0952381' to each side of the equation. -68.0952381 + 68.0952381 + a = 114.346214148 + 68.0952381 Combine like terms: -68.0952381 + 68.0952381 = 0.0000000 0.0000000 + a = 114.346214148 + 68.0952381 a = 114.346214148 + 68.0952381 Combine like terms: 114.346214148 + 68.0952381 = 182.441452248 a = 182.441452248 Simplifying a = 182.441452248Subproblem 2
a + -68.0952381 = -114.346214148 Simplifying a + -68.0952381 = -114.346214148 Reorder the terms: -68.0952381 + a = -114.346214148 Solving -68.0952381 + a = -114.346214148 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '68.0952381' to each side of the equation. -68.0952381 + 68.0952381 + a = -114.346214148 + 68.0952381 Combine like terms: -68.0952381 + 68.0952381 = 0.0000000 0.0000000 + a = -114.346214148 + 68.0952381 a = -114.346214148 + 68.0952381 Combine like terms: -114.346214148 + 68.0952381 = -46.250976048 a = -46.250976048 Simplifying a = -46.250976048Solution
The solution to the problem is based on the solutions from the subproblems. a = {182.441452248, -46.250976048}
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