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Simplifying (a2 + -4a + 6) + -1(2a2 + -1a + -7) = 0 Reorder the terms: (6 + -4a + a2) + -1(2a2 + -1a + -7) = 0 Remove parenthesis around (6 + -4a + a2) 6 + -4a + a2 + -1(2a2 + -1a + -7) = 0 Reorder the terms: 6 + -4a + a2 + -1(-7 + -1a + 2a2) = 0 6 + -4a + a2 + (-7 * -1 + -1a * -1 + 2a2 * -1) = 0 6 + -4a + a2 + (7 + 1a + -2a2) = 0 Reorder the terms: 6 + 7 + -4a + 1a + a2 + -2a2 = 0 Combine like terms: 6 + 7 = 13 13 + -4a + 1a + a2 + -2a2 = 0 Combine like terms: -4a + 1a = -3a 13 + -3a + a2 + -2a2 = 0 Combine like terms: a2 + -2a2 = -1a2 13 + -3a + -1a2 = 0 Solving 13 + -3a + -1a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -13 + 3a + a2 = 0 Move the constant term to the right: Add '13' to each side of the equation. -13 + 3a + 13 + a2 = 0 + 13 Reorder the terms: -13 + 13 + 3a + a2 = 0 + 13 Combine like terms: -13 + 13 = 0 0 + 3a + a2 = 0 + 13 3a + a2 = 0 + 13 Combine like terms: 0 + 13 = 13 3a + a2 = 13 The a term is 3a. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3a + 2.25 + a2 = 13 + 2.25 Reorder the terms: 2.25 + 3a + a2 = 13 + 2.25 Combine like terms: 13 + 2.25 = 15.25 2.25 + 3a + a2 = 15.25 Factor a perfect square on the left side: (a + 1.5)(a + 1.5) = 15.25 Calculate the square root of the right side: 3.905124838 Break this problem into two subproblems by setting (a + 1.5) equal to 3.905124838 and -3.905124838.Subproblem 1
a + 1.5 = 3.905124838 Simplifying a + 1.5 = 3.905124838 Reorder the terms: 1.5 + a = 3.905124838 Solving 1.5 + a = 3.905124838 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + a = 3.905124838 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + a = 3.905124838 + -1.5 a = 3.905124838 + -1.5 Combine like terms: 3.905124838 + -1.5 = 2.405124838 a = 2.405124838 Simplifying a = 2.405124838Subproblem 2
a + 1.5 = -3.905124838 Simplifying a + 1.5 = -3.905124838 Reorder the terms: 1.5 + a = -3.905124838 Solving 1.5 + a = -3.905124838 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + a = -3.905124838 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + a = -3.905124838 + -1.5 a = -3.905124838 + -1.5 Combine like terms: -3.905124838 + -1.5 = -5.405124838 a = -5.405124838 Simplifying a = -5.405124838Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.405124838, -5.405124838}
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