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Simplifying 4a2 + 5a + -14 = 0 Reorder the terms: -14 + 5a + 4a2 = 0 Solving -14 + 5a + 4a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -3.5 + 1.25a + a2 = 0 Move the constant term to the right: Add '3.5' to each side of the equation. -3.5 + 1.25a + 3.5 + a2 = 0 + 3.5 Reorder the terms: -3.5 + 3.5 + 1.25a + a2 = 0 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + 1.25a + a2 = 0 + 3.5 1.25a + a2 = 0 + 3.5 Combine like terms: 0 + 3.5 = 3.5 1.25a + a2 = 3.5 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 = 3.5 + 0.390625 Reorder the terms: 0.390625 + 1.25a + a2 = 3.5 + 0.390625 Combine like terms: 3.5 + 0.390625 = 3.890625 0.390625 + 1.25a + a2 = 3.890625 Factor a perfect square on the left side: (a + 0.625)(a + 0.625) = 3.890625 Calculate the square root of the right side: 1.97246673 Break this problem into two subproblems by setting (a + 0.625) equal to 1.97246673 and -1.97246673.Subproblem 1
a + 0.625 = 1.97246673 Simplifying a + 0.625 = 1.97246673 Reorder the terms: 0.625 + a = 1.97246673 Solving 0.625 + a = 1.97246673 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 = 1.97246673 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + a = 1.97246673 + -0.625 a = 1.97246673 + -0.625 Combine like terms: 1.97246673 + -0.625 = 1.34746673 a = 1.34746673 Simplifying a = 1.34746673Subproblem 2
a + 0.625 = -1.97246673 Simplifying a + 0.625 = -1.97246673 Reorder the terms: 0.625 + a = -1.97246673 Solving 0.625 + a = -1.97246673 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 = -1.97246673 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + a = -1.97246673 + -0.625 a = -1.97246673 + -0.625 Combine like terms: -1.97246673 + -0.625 = -2.59746673 a = -2.59746673 Simplifying a = -2.59746673Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.34746673, -2.59746673}
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