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Simplifying (4a2 + -3a) + -1(-1a2 + -2a + 1) = 0 Reorder the terms: (-3a + 4a2) + -1(-1a2 + -2a + 1) = 0 Remove parenthesis around (-3a + 4a2) -3a + 4a2 + -1(-1a2 + -2a + 1) = 0 Reorder the terms: -3a + 4a2 + -1(1 + -2a + -1a2) = 0 -3a + 4a2 + (1 * -1 + -2a * -1 + -1a2 * -1) = 0 -3a + 4a2 + (-1 + 2a + 1a2) = 0 Reorder the terms: -1 + -3a + 2a + 4a2 + 1a2 = 0 Combine like terms: -3a + 2a = -1a -1 + -1a + 4a2 + 1a2 = 0 Combine like terms: 4a2 + 1a2 = 5a2 -1 + -1a + 5a2 = 0 Solving -1 + -1a + 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'. -0.2 + -0.2a + a2 = 0 Move the constant term to the right: Add '0.2' to each side of the equation. -0.2 + -0.2a + 0.2 + a2 = 0 + 0.2 Reorder the terms: -0.2 + 0.2 + -0.2a + a2 = 0 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + -0.2a + a2 = 0 + 0.2 -0.2a + a2 = 0 + 0.2 Combine like terms: 0 + 0.2 = 0.2 -0.2a + a2 = 0.2 The a term is -0.2a. Take half its coefficient (-0.1). Square it (0.01) and add it to both sides. Add '0.01' to each side of the equation. -0.2a + 0.01 + a2 = 0.2 + 0.01 Reorder the terms: 0.01 + -0.2a + a2 = 0.2 + 0.01 Combine like terms: 0.2 + 0.01 = 0.21 0.01 + -0.2a + a2 = 0.21 Factor a perfect square on the left side: (a + -0.1)(a + -0.1) = 0.21 Calculate the square root of the right side: 0.458257569 Break this problem into two subproblems by setting (a + -0.1) equal to 0.458257569 and -0.458257569.Subproblem 1
a + -0.1 = 0.458257569 Simplifying a + -0.1 = 0.458257569 Reorder the terms: -0.1 + a = 0.458257569 Solving -0.1 + a = 0.458257569 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.1' to each side of the equation. -0.1 + 0.1 + a = 0.458257569 + 0.1 Combine like terms: -0.1 + 0.1 = 0.0 0.0 + a = 0.458257569 + 0.1 a = 0.458257569 + 0.1 Combine like terms: 0.458257569 + 0.1 = 0.558257569 a = 0.558257569 Simplifying a = 0.558257569Subproblem 2
a + -0.1 = -0.458257569 Simplifying a + -0.1 = -0.458257569 Reorder the terms: -0.1 + a = -0.458257569 Solving -0.1 + a = -0.458257569 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.1' to each side of the equation. -0.1 + 0.1 + a = -0.458257569 + 0.1 Combine like terms: -0.1 + 0.1 = 0.0 0.0 + a = -0.458257569 + 0.1 a = -0.458257569 + 0.1 Combine like terms: -0.458257569 + 0.1 = -0.358257569 a = -0.358257569 Simplifying a = -0.358257569Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.558257569, -0.358257569}
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