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Simplifying 10.125a2 + -19a + 7.5 = 0 Reorder the terms: 7.5 + -19a + 10.125a2 = 0 Solving 7.5 + -19a + 10.125a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 10.125 the coefficient of the squared term: Divide each side by '10.125'. 0.7407407407 + -1.87654321a + a2 = 0 Move the constant term to the right: Add '-0.7407407407' to each side of the equation. 0.7407407407 + -1.87654321a + -0.7407407407 + a2 = 0 + -0.7407407407 Reorder the terms: 0.7407407407 + -0.7407407407 + -1.87654321a + a2 = 0 + -0.7407407407 Combine like terms: 0.7407407407 + -0.7407407407 = 0.0000000000 0.0000000000 + -1.87654321a + a2 = 0 + -0.7407407407 -1.87654321a + a2 = 0 + -0.7407407407 Combine like terms: 0 + -0.7407407407 = -0.7407407407 -1.87654321a + a2 = -0.7407407407 The a term is -1.87654321a. Take half its coefficient (-0.938271605). Square it (0.8803536047) and add it to both sides. Add '0.8803536047' to each side of the equation. -1.87654321a + 0.8803536047 + a2 = -0.7407407407 + 0.8803536047 Reorder the terms: 0.8803536047 + -1.87654321a + a2 = -0.7407407407 + 0.8803536047 Combine like terms: -0.7407407407 + 0.8803536047 = 0.139612864 0.8803536047 + -1.87654321a + a2 = 0.139612864 Factor a perfect square on the left side: (a + -0.938271605)(a + -0.938271605) = 0.139612864 Calculate the square root of the right side: 0.373648048 Break this problem into two subproblems by setting (a + -0.938271605) equal to 0.373648048 and -0.373648048.Subproblem 1
a + -0.938271605 = 0.373648048 Simplifying a + -0.938271605 = 0.373648048 Reorder the terms: -0.938271605 + a = 0.373648048 Solving -0.938271605 + a = 0.373648048 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.938271605' to each side of the equation. -0.938271605 + 0.938271605 + a = 0.373648048 + 0.938271605 Combine like terms: -0.938271605 + 0.938271605 = 0.000000000 0.000000000 + a = 0.373648048 + 0.938271605 a = 0.373648048 + 0.938271605 Combine like terms: 0.373648048 + 0.938271605 = 1.311919653 a = 1.311919653 Simplifying a = 1.311919653Subproblem 2
a + -0.938271605 = -0.373648048 Simplifying a + -0.938271605 = -0.373648048 Reorder the terms: -0.938271605 + a = -0.373648048 Solving -0.938271605 + a = -0.373648048 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.938271605' to each side of the equation. -0.938271605 + 0.938271605 + a = -0.373648048 + 0.938271605 Combine like terms: -0.938271605 + 0.938271605 = 0.000000000 0.000000000 + a = -0.373648048 + 0.938271605 a = -0.373648048 + 0.938271605 Combine like terms: -0.373648048 + 0.938271605 = 0.564623557 a = 0.564623557 Simplifying a = 0.564623557Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.311919653, 0.564623557}
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