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Simplifying 2a2 + 15a + -45 = 0 Reorder the terms: -45 + 15a + 2a2 = 0 Solving -45 + 15a + 2a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -22.5 + 7.5a + a2 = 0 Move the constant term to the right: Add '22.5' to each side of the equation. -22.5 + 7.5a + 22.5 + a2 = 0 + 22.5 Reorder the terms: -22.5 + 22.5 + 7.5a + a2 = 0 + 22.5 Combine like terms: -22.5 + 22.5 = 0.0 0.0 + 7.5a + a2 = 0 + 22.5 7.5a + a2 = 0 + 22.5 Combine like terms: 0 + 22.5 = 22.5 7.5a + a2 = 22.5 The a term is 7.5a. Take half its coefficient (3.75). Square it (14.0625) and add it to both sides. Add '14.0625' to each side of the equation. 7.5a + 14.0625 + a2 = 22.5 + 14.0625 Reorder the terms: 14.0625 + 7.5a + a2 = 22.5 + 14.0625 Combine like terms: 22.5 + 14.0625 = 36.5625 14.0625 + 7.5a + a2 = 36.5625 Factor a perfect square on the left side: (a + 3.75)(a + 3.75) = 36.5625 Calculate the square root of the right side: 6.046693311 Break this problem into two subproblems by setting (a + 3.75) equal to 6.046693311 and -6.046693311.Subproblem 1
a + 3.75 = 6.046693311 Simplifying a + 3.75 = 6.046693311 Reorder the terms: 3.75 + a = 6.046693311 Solving 3.75 + a = 6.046693311 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-3.75' to each side of the equation. 3.75 + -3.75 + a = 6.046693311 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + a = 6.046693311 + -3.75 a = 6.046693311 + -3.75 Combine like terms: 6.046693311 + -3.75 = 2.296693311 a = 2.296693311 Simplifying a = 2.296693311Subproblem 2
a + 3.75 = -6.046693311 Simplifying a + 3.75 = -6.046693311 Reorder the terms: 3.75 + a = -6.046693311 Solving 3.75 + a = -6.046693311 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-3.75' to each side of the equation. 3.75 + -3.75 + a = -6.046693311 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + a = -6.046693311 + -3.75 a = -6.046693311 + -3.75 Combine like terms: -6.046693311 + -3.75 = -9.796693311 a = -9.796693311 Simplifying a = -9.796693311Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.296693311, -9.796693311}
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