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Simplifying 6a2 + -24a + 19 = 0 Reorder the terms: 19 + -24a + 6a2 = 0 Solving 19 + -24a + 6a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. 3.166666667 + -4a + a2 = 0 Move the constant term to the right: Add '-3.166666667' to each side of the equation. 3.166666667 + -4a + -3.166666667 + a2 = 0 + -3.166666667 Reorder the terms: 3.166666667 + -3.166666667 + -4a + a2 = 0 + -3.166666667 Combine like terms: 3.166666667 + -3.166666667 = 0.000000000 0.000000000 + -4a + a2 = 0 + -3.166666667 -4a + a2 = 0 + -3.166666667 Combine like terms: 0 + -3.166666667 = -3.166666667 -4a + a2 = -3.166666667 The a term is -4a. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4a + 4 + a2 = -3.166666667 + 4 Reorder the terms: 4 + -4a + a2 = -3.166666667 + 4 Combine like terms: -3.166666667 + 4 = 0.833333333 4 + -4a + a2 = 0.833333333 Factor a perfect square on the left side: (a + -2)(a + -2) = 0.833333333 Calculate the square root of the right side: 0.912870929 Break this problem into two subproblems by setting (a + -2) equal to 0.912870929 and -0.912870929.Subproblem 1
a + -2 = 0.912870929 Simplifying a + -2 = 0.912870929 Reorder the terms: -2 + a = 0.912870929 Solving -2 + a = 0.912870929 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + a = 0.912870929 + 2 Combine like terms: -2 + 2 = 0 0 + a = 0.912870929 + 2 a = 0.912870929 + 2 Combine like terms: 0.912870929 + 2 = 2.912870929 a = 2.912870929 Simplifying a = 2.912870929Subproblem 2
a + -2 = -0.912870929 Simplifying a + -2 = -0.912870929 Reorder the terms: -2 + a = -0.912870929 Solving -2 + a = -0.912870929 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + a = -0.912870929 + 2 Combine like terms: -2 + 2 = 0 0 + a = -0.912870929 + 2 a = -0.912870929 + 2 Combine like terms: -0.912870929 + 2 = 1.087129071 a = 1.087129071 Simplifying a = 1.087129071Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.912870929, 1.087129071}
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