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Simplifying 5a2 + -41a + 1 = 7 Reorder the terms: 1 + -41a + 5a2 = 7 Solving 1 + -41a + 5a2 = 7 Solving for variable 'a'. Reorder the terms: 1 + -7 + -41a + 5a2 = 7 + -7 Combine like terms: 1 + -7 = -6 -6 + -41a + 5a2 = 7 + -7 Combine like terms: 7 + -7 = 0 -6 + -41a + 5a2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -1.2 + -8.2a + a2 = 0 Move the constant term to the right: Add '1.2' to each side of the equation. -1.2 + -8.2a + 1.2 + a2 = 0 + 1.2 Reorder the terms: -1.2 + 1.2 + -8.2a + a2 = 0 + 1.2 Combine like terms: -1.2 + 1.2 = 0.0 0.0 + -8.2a + a2 = 0 + 1.2 -8.2a + a2 = 0 + 1.2 Combine like terms: 0 + 1.2 = 1.2 -8.2a + a2 = 1.2 The a term is -8.2a. Take half its coefficient (-4.1). Square it (16.81) and add it to both sides. Add '16.81' to each side of the equation. -8.2a + 16.81 + a2 = 1.2 + 16.81 Reorder the terms: 16.81 + -8.2a + a2 = 1.2 + 16.81 Combine like terms: 1.2 + 16.81 = 18.01 16.81 + -8.2a + a2 = 18.01 Factor a perfect square on the left side: (a + -4.1)(a + -4.1) = 18.01 Calculate the square root of the right side: 4.243819035 Break this problem into two subproblems by setting (a + -4.1) equal to 4.243819035 and -4.243819035.Subproblem 1
a + -4.1 = 4.243819035 Simplifying a + -4.1 = 4.243819035 Reorder the terms: -4.1 + a = 4.243819035 Solving -4.1 + a = 4.243819035 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '4.1' to each side of the equation. -4.1 + 4.1 + a = 4.243819035 + 4.1 Combine like terms: -4.1 + 4.1 = 0.0 0.0 + a = 4.243819035 + 4.1 a = 4.243819035 + 4.1 Combine like terms: 4.243819035 + 4.1 = 8.343819035 a = 8.343819035 Simplifying a = 8.343819035Subproblem 2
a + -4.1 = -4.243819035 Simplifying a + -4.1 = -4.243819035 Reorder the terms: -4.1 + a = -4.243819035 Solving -4.1 + a = -4.243819035 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '4.1' to each side of the equation. -4.1 + 4.1 + a = -4.243819035 + 4.1 Combine like terms: -4.1 + 4.1 = 0.0 0.0 + a = -4.243819035 + 4.1 a = -4.243819035 + 4.1 Combine like terms: -4.243819035 + 4.1 = -0.143819035 a = -0.143819035 Simplifying a = -0.143819035Solution
The solution to the problem is based on the solutions from the subproblems. a = {8.343819035, -0.143819035}
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