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Simplifying 10a * 10a + 7a + -12 = 0 Reorder the terms for easier multiplication: 10 * 10a * a + 7a + -12 = 0 Multiply 10 * 10 100a * a + 7a + -12 = 0 Multiply a * a 100a2 + 7a + -12 = 0 Reorder the terms: -12 + 7a + 100a2 = 0 Solving -12 + 7a + 100a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 100 the coefficient of the squared term: Divide each side by '100'. -0.12 + 0.07a + a2 = 0.00 Move the constant term to the right: Add '0.12' to each side of the equation. -0.12 + 0.07a + 0.12 + a2 = 0.00 + 0.12 Reorder the terms: -0.12 + 0.12 + 0.07a + a2 = 0.00 + 0.12 Combine like terms: -0.12 + 0.12 = 0.00 0.00 + 0.07a + a2 = 0.00 + 0.12 0.07a + a2 = 0.00 + 0.12 Combine like terms: 0.00 + 0.12 = 0.12 0.07a + a2 = 0.12 The a term is 0.07a. Take half its coefficient (0.035). Square it (0.001225) and add it to both sides. Add '0.001225' to each side of the equation. 0.07a + 0.001225 + a2 = 0.12 + 0.001225 Reorder the terms: 0.001225 + 0.07a + a2 = 0.12 + 0.001225 Combine like terms: 0.12 + 0.001225 = 0.121225 0.001225 + 0.07a + a2 = 0.121225 Factor a perfect square on the left side: (a + 0.035)(a + 0.035) = 0.121225 Calculate the square root of the right side: 0.348173807 Break this problem into two subproblems by setting (a + 0.035) equal to 0.348173807 and -0.348173807.Subproblem 1
a + 0.035 = 0.348173807 Simplifying a + 0.035 = 0.348173807 Reorder the terms: 0.035 + a = 0.348173807 Solving 0.035 + a = 0.348173807 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.035' to each side of the equation. 0.035 + -0.035 + a = 0.348173807 + -0.035 Combine like terms: 0.035 + -0.035 = 0.000 0.000 + a = 0.348173807 + -0.035 a = 0.348173807 + -0.035 Combine like terms: 0.348173807 + -0.035 = 0.313173807 a = 0.313173807 Simplifying a = 0.313173807Subproblem 2
a + 0.035 = -0.348173807 Simplifying a + 0.035 = -0.348173807 Reorder the terms: 0.035 + a = -0.348173807 Solving 0.035 + a = -0.348173807 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.035' to each side of the equation. 0.035 + -0.035 + a = -0.348173807 + -0.035 Combine like terms: 0.035 + -0.035 = 0.000 0.000 + a = -0.348173807 + -0.035 a = -0.348173807 + -0.035 Combine like terms: -0.348173807 + -0.035 = -0.383173807 a = -0.383173807 Simplifying a = -0.383173807Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.313173807, -0.383173807}
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