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Simplifying 5a2 + -14a + -3 = 10 Reorder the terms: -3 + -14a + 5a2 = 10 Solving -3 + -14a + 5a2 = 10 Solving for variable 'a'. Reorder the terms: -3 + -10 + -14a + 5a2 = 10 + -10 Combine like terms: -3 + -10 = -13 -13 + -14a + 5a2 = 10 + -10 Combine like terms: 10 + -10 = 0 -13 + -14a + 5a2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -2.6 + -2.8a + a2 = 0 Move the constant term to the right: Add '2.6' to each side of the equation. -2.6 + -2.8a + 2.6 + a2 = 0 + 2.6 Reorder the terms: -2.6 + 2.6 + -2.8a + a2 = 0 + 2.6 Combine like terms: -2.6 + 2.6 = 0.0 0.0 + -2.8a + a2 = 0 + 2.6 -2.8a + a2 = 0 + 2.6 Combine like terms: 0 + 2.6 = 2.6 -2.8a + a2 = 2.6 The a term is -2.8a. Take half its coefficient (-1.4). Square it (1.96) and add it to both sides. Add '1.96' to each side of the equation. -2.8a + 1.96 + a2 = 2.6 + 1.96 Reorder the terms: 1.96 + -2.8a + a2 = 2.6 + 1.96 Combine like terms: 2.6 + 1.96 = 4.56 1.96 + -2.8a + a2 = 4.56 Factor a perfect square on the left side: (a + -1.4)(a + -1.4) = 4.56 Calculate the square root of the right side: 2.13541565 Break this problem into two subproblems by setting (a + -1.4) equal to 2.13541565 and -2.13541565.Subproblem 1
a + -1.4 = 2.13541565 Simplifying a + -1.4 = 2.13541565 Reorder the terms: -1.4 + a = 2.13541565 Solving -1.4 + a = 2.13541565 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.4' to each side of the equation. -1.4 + 1.4 + a = 2.13541565 + 1.4 Combine like terms: -1.4 + 1.4 = 0.0 0.0 + a = 2.13541565 + 1.4 a = 2.13541565 + 1.4 Combine like terms: 2.13541565 + 1.4 = 3.53541565 a = 3.53541565 Simplifying a = 3.53541565Subproblem 2
a + -1.4 = -2.13541565 Simplifying a + -1.4 = -2.13541565 Reorder the terms: -1.4 + a = -2.13541565 Solving -1.4 + a = -2.13541565 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.4' to each side of the equation. -1.4 + 1.4 + a = -2.13541565 + 1.4 Combine like terms: -1.4 + 1.4 = 0.0 0.0 + a = -2.13541565 + 1.4 a = -2.13541565 + 1.4 Combine like terms: -2.13541565 + 1.4 = -0.73541565 a = -0.73541565 Simplifying a = -0.73541565Solution
The solution to the problem is based on the solutions from the subproblems. a = {3.53541565, -0.73541565}
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