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Simplifying 6a2 + 14a + 3 = 0 Reorder the terms: 3 + 14a + 6a2 = 0 Solving 3 + 14a + 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'. 0.5 + 2.333333333a + a2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + 2.333333333a + -0.5 + a2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + 2.333333333a + a2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + 2.333333333a + a2 = 0 + -0.5 2.333333333a + a2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 2.333333333a + a2 = -0.5 The a term is 2.333333333a. Take half its coefficient (1.166666667). Square it (1.361111112) and add it to both sides. Add '1.361111112' to each side of the equation. 2.333333333a + 1.361111112 + a2 = -0.5 + 1.361111112 Reorder the terms: 1.361111112 + 2.333333333a + a2 = -0.5 + 1.361111112 Combine like terms: -0.5 + 1.361111112 = 0.861111112 1.361111112 + 2.333333333a + a2 = 0.861111112 Factor a perfect square on the left side: (a + 1.166666667)(a + 1.166666667) = 0.861111112 Calculate the square root of the right side: 0.927960728 Break this problem into two subproblems by setting (a + 1.166666667) equal to 0.927960728 and -0.927960728.Subproblem 1
a + 1.166666667 = 0.927960728 Simplifying a + 1.166666667 = 0.927960728 Reorder the terms: 1.166666667 + a = 0.927960728 Solving 1.166666667 + a = 0.927960728 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.166666667' to each side of the equation. 1.166666667 + -1.166666667 + a = 0.927960728 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + a = 0.927960728 + -1.166666667 a = 0.927960728 + -1.166666667 Combine like terms: 0.927960728 + -1.166666667 = -0.238705939 a = -0.238705939 Simplifying a = -0.238705939Subproblem 2
a + 1.166666667 = -0.927960728 Simplifying a + 1.166666667 = -0.927960728 Reorder the terms: 1.166666667 + a = -0.927960728 Solving 1.166666667 + a = -0.927960728 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.166666667' to each side of the equation. 1.166666667 + -1.166666667 + a = -0.927960728 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + a = -0.927960728 + -1.166666667 a = -0.927960728 + -1.166666667 Combine like terms: -0.927960728 + -1.166666667 = -2.094627395 a = -2.094627395 Simplifying a = -2.094627395Solution
The solution to the problem is based on the solutions from the subproblems. a = {-0.238705939, -2.094627395}
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