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Simplifying 5a2 + 8a + -23 = 0 Reorder the terms: -23 + 8a + 5a2 = 0 Solving -23 + 8a + 5a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -4.6 + 1.6a + a2 = 0 Move the constant term to the right: Add '4.6' to each side of the equation. -4.6 + 1.6a + 4.6 + a2 = 0 + 4.6 Reorder the terms: -4.6 + 4.6 + 1.6a + a2 = 0 + 4.6 Combine like terms: -4.6 + 4.6 = 0.0 0.0 + 1.6a + a2 = 0 + 4.6 1.6a + a2 = 0 + 4.6 Combine like terms: 0 + 4.6 = 4.6 1.6a + a2 = 4.6 The a term is 1.6a. Take half its coefficient (0.8). Square it (0.64) and add it to both sides. Add '0.64' to each side of the equation. 1.6a + 0.64 + a2 = 4.6 + 0.64 Reorder the terms: 0.64 + 1.6a + a2 = 4.6 + 0.64 Combine like terms: 4.6 + 0.64 = 5.24 0.64 + 1.6a + a2 = 5.24 Factor a perfect square on the left side: (a + 0.8)(a + 0.8) = 5.24 Calculate the square root of the right side: 2.289104628 Break this problem into two subproblems by setting (a + 0.8) equal to 2.289104628 and -2.289104628.Subproblem 1
a + 0.8 = 2.289104628 Simplifying a + 0.8 = 2.289104628 Reorder the terms: 0.8 + a = 2.289104628 Solving 0.8 + a = 2.289104628 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.8' to each side of the equation. 0.8 + -0.8 + a = 2.289104628 + -0.8 Combine like terms: 0.8 + -0.8 = 0.0 0.0 + a = 2.289104628 + -0.8 a = 2.289104628 + -0.8 Combine like terms: 2.289104628 + -0.8 = 1.489104628 a = 1.489104628 Simplifying a = 1.489104628Subproblem 2
a + 0.8 = -2.289104628 Simplifying a + 0.8 = -2.289104628 Reorder the terms: 0.8 + a = -2.289104628 Solving 0.8 + a = -2.289104628 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.8' to each side of the equation. 0.8 + -0.8 + a = -2.289104628 + -0.8 Combine like terms: 0.8 + -0.8 = 0.0 0.0 + a = -2.289104628 + -0.8 a = -2.289104628 + -0.8 Combine like terms: -2.289104628 + -0.8 = -3.089104628 a = -3.089104628 Simplifying a = -3.089104628Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.489104628, -3.089104628}
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