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Simplifying 2a2 + 3a + -7 = 90 Reorder the terms: -7 + 3a + 2a2 = 90 Solving -7 + 3a + 2a2 = 90 Solving for variable 'a'. Reorder the terms: -7 + -90 + 3a + 2a2 = 90 + -90 Combine like terms: -7 + -90 = -97 -97 + 3a + 2a2 = 90 + -90 Combine like terms: 90 + -90 = 0 -97 + 3a + 2a2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -48.5 + 1.5a + a2 = 0 Move the constant term to the right: Add '48.5' to each side of the equation. -48.5 + 1.5a + 48.5 + a2 = 0 + 48.5 Reorder the terms: -48.5 + 48.5 + 1.5a + a2 = 0 + 48.5 Combine like terms: -48.5 + 48.5 = 0.0 0.0 + 1.5a + a2 = 0 + 48.5 1.5a + a2 = 0 + 48.5 Combine like terms: 0 + 48.5 = 48.5 1.5a + a2 = 48.5 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 = 48.5 + 0.5625 Reorder the terms: 0.5625 + 1.5a + a2 = 48.5 + 0.5625 Combine like terms: 48.5 + 0.5625 = 49.0625 0.5625 + 1.5a + a2 = 49.0625 Factor a perfect square on the left side: (a + 0.75)(a + 0.75) = 49.0625 Calculate the square root of the right side: 7.004462863 Break this problem into two subproblems by setting (a + 0.75) equal to 7.004462863 and -7.004462863.Subproblem 1
a + 0.75 = 7.004462863 Simplifying a + 0.75 = 7.004462863 Reorder the terms: 0.75 + a = 7.004462863 Solving 0.75 + a = 7.004462863 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 = 7.004462863 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + a = 7.004462863 + -0.75 a = 7.004462863 + -0.75 Combine like terms: 7.004462863 + -0.75 = 6.254462863 a = 6.254462863 Simplifying a = 6.254462863Subproblem 2
a + 0.75 = -7.004462863 Simplifying a + 0.75 = -7.004462863 Reorder the terms: 0.75 + a = -7.004462863 Solving 0.75 + a = -7.004462863 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 = -7.004462863 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + a = -7.004462863 + -0.75 a = -7.004462863 + -0.75 Combine like terms: -7.004462863 + -0.75 = -7.754462863 a = -7.754462863 Simplifying a = -7.754462863Solution
The solution to the problem is based on the solutions from the subproblems. a = {6.254462863, -7.754462863}
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