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Simplifying 5(a + 6) + -3a(a + 4) = 0 Reorder the terms: 5(6 + a) + -3a(a + 4) = 0 (6 * 5 + a * 5) + -3a(a + 4) = 0 (30 + 5a) + -3a(a + 4) = 0 Reorder the terms: 30 + 5a + -3a(4 + a) = 0 30 + 5a + (4 * -3a + a * -3a) = 0 30 + 5a + (-12a + -3a2) = 0 Combine like terms: 5a + -12a = -7a 30 + -7a + -3a2 = 0 Solving 30 + -7a + -3a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -10 + 2.333333333a + a2 = 0 Move the constant term to the right: Add '10' to each side of the equation. -10 + 2.333333333a + 10 + a2 = 0 + 10 Reorder the terms: -10 + 10 + 2.333333333a + a2 = 0 + 10 Combine like terms: -10 + 10 = 0 0 + 2.333333333a + a2 = 0 + 10 2.333333333a + a2 = 0 + 10 Combine like terms: 0 + 10 = 10 2.333333333a + a2 = 10 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 = 10 + 1.361111112 Reorder the terms: 1.361111112 + 2.333333333a + a2 = 10 + 1.361111112 Combine like terms: 10 + 1.361111112 = 11.361111112 1.361111112 + 2.333333333a + a2 = 11.361111112 Factor a perfect square on the left side: (a + 1.166666667)(a + 1.166666667) = 11.361111112 Calculate the square root of the right side: 3.370624736 Break this problem into two subproblems by setting (a + 1.166666667) equal to 3.370624736 and -3.370624736.Subproblem 1
a + 1.166666667 = 3.370624736 Simplifying a + 1.166666667 = 3.370624736 Reorder the terms: 1.166666667 + a = 3.370624736 Solving 1.166666667 + a = 3.370624736 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 = 3.370624736 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + a = 3.370624736 + -1.166666667 a = 3.370624736 + -1.166666667 Combine like terms: 3.370624736 + -1.166666667 = 2.203958069 a = 2.203958069 Simplifying a = 2.203958069Subproblem 2
a + 1.166666667 = -3.370624736 Simplifying a + 1.166666667 = -3.370624736 Reorder the terms: 1.166666667 + a = -3.370624736 Solving 1.166666667 + a = -3.370624736 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 = -3.370624736 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + a = -3.370624736 + -1.166666667 a = -3.370624736 + -1.166666667 Combine like terms: -3.370624736 + -1.166666667 = -4.537291403 a = -4.537291403 Simplifying a = -4.537291403Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.203958069, -4.537291403}
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