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Simplifying 7a2 + 25a + 21 = 0 Reorder the terms: 21 + 25a + 7a2 = 0 Solving 21 + 25a + 7a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. 3 + 3.571428571a + a2 = 0 Move the constant term to the right: Add '-3' to each side of the equation. 3 + 3.571428571a + -3 + a2 = 0 + -3 Reorder the terms: 3 + -3 + 3.571428571a + a2 = 0 + -3 Combine like terms: 3 + -3 = 0 0 + 3.571428571a + a2 = 0 + -3 3.571428571a + a2 = 0 + -3 Combine like terms: 0 + -3 = -3 3.571428571a + a2 = -3 The a term is 3.571428571a. Take half its coefficient (1.785714286). Square it (3.188775511) and add it to both sides. Add '3.188775511' to each side of the equation. 3.571428571a + 3.188775511 + a2 = -3 + 3.188775511 Reorder the terms: 3.188775511 + 3.571428571a + a2 = -3 + 3.188775511 Combine like terms: -3 + 3.188775511 = 0.188775511 3.188775511 + 3.571428571a + a2 = 0.188775511 Factor a perfect square on the left side: (a + 1.785714286)(a + 1.785714286) = 0.188775511 Calculate the square root of the right side: 0.434483039 Break this problem into two subproblems by setting (a + 1.785714286) equal to 0.434483039 and -0.434483039.Subproblem 1
a + 1.785714286 = 0.434483039 Simplifying a + 1.785714286 = 0.434483039 Reorder the terms: 1.785714286 + a = 0.434483039 Solving 1.785714286 + a = 0.434483039 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.785714286' to each side of the equation. 1.785714286 + -1.785714286 + a = 0.434483039 + -1.785714286 Combine like terms: 1.785714286 + -1.785714286 = 0.000000000 0.000000000 + a = 0.434483039 + -1.785714286 a = 0.434483039 + -1.785714286 Combine like terms: 0.434483039 + -1.785714286 = -1.351231247 a = -1.351231247 Simplifying a = -1.351231247Subproblem 2
a + 1.785714286 = -0.434483039 Simplifying a + 1.785714286 = -0.434483039 Reorder the terms: 1.785714286 + a = -0.434483039 Solving 1.785714286 + a = -0.434483039 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.785714286' to each side of the equation. 1.785714286 + -1.785714286 + a = -0.434483039 + -1.785714286 Combine like terms: 1.785714286 + -1.785714286 = 0.000000000 0.000000000 + a = -0.434483039 + -1.785714286 a = -0.434483039 + -1.785714286 Combine like terms: -0.434483039 + -1.785714286 = -2.220197325 a = -2.220197325 Simplifying a = -2.220197325Solution
The solution to the problem is based on the solutions from the subproblems. a = {-1.351231247, -2.220197325}
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