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Simplifying 2a2 + 3a + -7 = 50 Reorder the terms: -7 + 3a + 2a2 = 50 Solving -7 + 3a + 2a2 = 50 Solving for variable 'a'. Reorder the terms: -7 + -50 + 3a + 2a2 = 50 + -50 Combine like terms: -7 + -50 = -57 -57 + 3a + 2a2 = 50 + -50 Combine like terms: 50 + -50 = 0 -57 + 3a + 2a2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -28.5 + 1.5a + a2 = 0 Move the constant term to the right: Add '28.5' to each side of the equation. -28.5 + 1.5a + 28.5 + a2 = 0 + 28.5 Reorder the terms: -28.5 + 28.5 + 1.5a + a2 = 0 + 28.5 Combine like terms: -28.5 + 28.5 = 0.0 0.0 + 1.5a + a2 = 0 + 28.5 1.5a + a2 = 0 + 28.5 Combine like terms: 0 + 28.5 = 28.5 1.5a + a2 = 28.5 The a term is 1.5a. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5a + 0.5625 + a2 = 28.5 + 0.5625 Reorder the terms: 0.5625 + 1.5a + a2 = 28.5 + 0.5625 Combine like terms: 28.5 + 0.5625 = 29.0625 0.5625 + 1.5a + a2 = 29.0625 Factor a perfect square on the left side: (a + 0.75)(a + 0.75) = 29.0625 Calculate the square root of the right side: 5.390964663 Break this problem into two subproblems by setting (a + 0.75) equal to 5.390964663 and -5.390964663.Subproblem 1
a + 0.75 = 5.390964663 Simplifying a + 0.75 = 5.390964663 Reorder the terms: 0.75 + a = 5.390964663 Solving 0.75 + a = 5.390964663 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + a = 5.390964663 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + a = 5.390964663 + -0.75 a = 5.390964663 + -0.75 Combine like terms: 5.390964663 + -0.75 = 4.640964663 a = 4.640964663 Simplifying a = 4.640964663Subproblem 2
a + 0.75 = -5.390964663 Simplifying a + 0.75 = -5.390964663 Reorder the terms: 0.75 + a = -5.390964663 Solving 0.75 + a = -5.390964663 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + a = -5.390964663 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + a = -5.390964663 + -0.75 a = -5.390964663 + -0.75 Combine like terms: -5.390964663 + -0.75 = -6.140964663 a = -6.140964663 Simplifying a = -6.140964663Solution
The solution to the problem is based on the solutions from the subproblems. a = {4.640964663, -6.140964663}
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