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Simplifying 18a2 = -57a + 35 Reorder the terms: 18a2 = 35 + -57a Solving 18a2 = 35 + -57a Solving for variable 'a'. Reorder the terms: -35 + 57a + 18a2 = 35 + -57a + -35 + 57a Reorder the terms: -35 + 57a + 18a2 = 35 + -35 + -57a + 57a Combine like terms: 35 + -35 = 0 -35 + 57a + 18a2 = 0 + -57a + 57a -35 + 57a + 18a2 = -57a + 57a Combine like terms: -57a + 57a = 0 -35 + 57a + 18a2 = 0 Begin completing the square. Divide all terms by 18 the coefficient of the squared term: Divide each side by '18'. -1.944444444 + 3.166666667a + a2 = 0 Move the constant term to the right: Add '1.944444444' to each side of the equation. -1.944444444 + 3.166666667a + 1.944444444 + a2 = 0 + 1.944444444 Reorder the terms: -1.944444444 + 1.944444444 + 3.166666667a + a2 = 0 + 1.944444444 Combine like terms: -1.944444444 + 1.944444444 = 0.000000000 0.000000000 + 3.166666667a + a2 = 0 + 1.944444444 3.166666667a + a2 = 0 + 1.944444444 Combine like terms: 0 + 1.944444444 = 1.944444444 3.166666667a + a2 = 1.944444444 The a term is 3.166666667a. Take half its coefficient (1.583333334). Square it (2.506944447) and add it to both sides. Add '2.506944447' to each side of the equation. 3.166666667a + 2.506944447 + a2 = 1.944444444 + 2.506944447 Reorder the terms: 2.506944447 + 3.166666667a + a2 = 1.944444444 + 2.506944447 Combine like terms: 1.944444444 + 2.506944447 = 4.451388891 2.506944447 + 3.166666667a + a2 = 4.451388891 Factor a perfect square on the left side: (a + 1.583333334)(a + 1.583333334) = 4.451388891 Calculate the square root of the right side: 2.109831484 Break this problem into two subproblems by setting (a + 1.583333334) equal to 2.109831484 and -2.109831484.Subproblem 1
a + 1.583333334 = 2.109831484 Simplifying a + 1.583333334 = 2.109831484 Reorder the terms: 1.583333334 + a = 2.109831484 Solving 1.583333334 + a = 2.109831484 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.583333334' to each side of the equation. 1.583333334 + -1.583333334 + a = 2.109831484 + -1.583333334 Combine like terms: 1.583333334 + -1.583333334 = 0.000000000 0.000000000 + a = 2.109831484 + -1.583333334 a = 2.109831484 + -1.583333334 Combine like terms: 2.109831484 + -1.583333334 = 0.52649815 a = 0.52649815 Simplifying a = 0.52649815Subproblem 2
a + 1.583333334 = -2.109831484 Simplifying a + 1.583333334 = -2.109831484 Reorder the terms: 1.583333334 + a = -2.109831484 Solving 1.583333334 + a = -2.109831484 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.583333334' to each side of the equation. 1.583333334 + -1.583333334 + a = -2.109831484 + -1.583333334 Combine like terms: 1.583333334 + -1.583333334 = 0.000000000 0.000000000 + a = -2.109831484 + -1.583333334 a = -2.109831484 + -1.583333334 Combine like terms: -2.109831484 + -1.583333334 = -3.693164818 a = -3.693164818 Simplifying a = -3.693164818Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.52649815, -3.693164818}
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