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Simplifying a2 + 5a + -3 + -7a + 6a2 + -4 = 0 Reorder the terms: -3 + -4 + 5a + -7a + a2 + 6a2 = 0 Combine like terms: -3 + -4 = -7 -7 + 5a + -7a + a2 + 6a2 = 0 Combine like terms: 5a + -7a = -2a -7 + -2a + a2 + 6a2 = 0 Combine like terms: a2 + 6a2 = 7a2 -7 + -2a + 7a2 = 0 Solving -7 + -2a + 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'. -1 + -0.2857142857a + a2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -0.2857142857a + 1 + a2 = 0 + 1 Reorder the terms: -1 + 1 + -0.2857142857a + a2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -0.2857142857a + a2 = 0 + 1 -0.2857142857a + a2 = 0 + 1 Combine like terms: 0 + 1 = 1 -0.2857142857a + a2 = 1 The a term is -0.2857142857a. Take half its coefficient (-0.1428571429). Square it (0.02040816328) and add it to both sides. Add '0.02040816328' to each side of the equation. -0.2857142857a + 0.02040816328 + a2 = 1 + 0.02040816328 Reorder the terms: 0.02040816328 + -0.2857142857a + a2 = 1 + 0.02040816328 Combine like terms: 1 + 0.02040816328 = 1.02040816328 0.02040816328 + -0.2857142857a + a2 = 1.02040816328 Factor a perfect square on the left side: (a + -0.1428571429)(a + -0.1428571429) = 1.02040816328 Calculate the square root of the right side: 1.010152545 Break this problem into two subproblems by setting (a + -0.1428571429) equal to 1.010152545 and -1.010152545.Subproblem 1
a + -0.1428571429 = 1.010152545 Simplifying a + -0.1428571429 = 1.010152545 Reorder the terms: -0.1428571429 + a = 1.010152545 Solving -0.1428571429 + a = 1.010152545 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.1428571429' to each side of the equation. -0.1428571429 + 0.1428571429 + a = 1.010152545 + 0.1428571429 Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000 0.0000000000 + a = 1.010152545 + 0.1428571429 a = 1.010152545 + 0.1428571429 Combine like terms: 1.010152545 + 0.1428571429 = 1.1530096879 a = 1.1530096879 Simplifying a = 1.1530096879Subproblem 2
a + -0.1428571429 = -1.010152545 Simplifying a + -0.1428571429 = -1.010152545 Reorder the terms: -0.1428571429 + a = -1.010152545 Solving -0.1428571429 + a = -1.010152545 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.1428571429' to each side of the equation. -0.1428571429 + 0.1428571429 + a = -1.010152545 + 0.1428571429 Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000 0.0000000000 + a = -1.010152545 + 0.1428571429 a = -1.010152545 + 0.1428571429 Combine like terms: -1.010152545 + 0.1428571429 = -0.8672954021 a = -0.8672954021 Simplifying a = -0.8672954021Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.1530096879, -0.8672954021}
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