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Simplifying 0 = (0.75 + -0.25a)(0.5)(0.25 + -0.25a) Reorder the terms for easier multiplication: 0 = 0.5(0.75 + -0.25a)(0.25 + -0.25a) Multiply (0.75 + -0.25a) * (0.25 + -0.25a) 0 = 0.5(0.75(0.25 + -0.25a) + -0.25a * (0.25 + -0.25a)) 0 = 0.5((0.25 * 0.75 + -0.25a * 0.75) + -0.25a * (0.25 + -0.25a)) 0 = 0.5((0.1875 + -0.1875a) + -0.25a * (0.25 + -0.25a)) 0 = 0.5(0.1875 + -0.1875a + (0.25 * -0.25a + -0.25a * -0.25a)) 0 = 0.5(0.1875 + -0.1875a + (-0.0625a + 0.0625a2)) Combine like terms: -0.1875a + -0.0625a = -0.25a 0 = 0.5(0.1875 + -0.25a + 0.0625a2) 0 = (0.1875 * 0.5 + -0.25a * 0.5 + 0.0625a2 * 0.5) 0 = (0.09375 + -0.125a + 0.03125a2) Solving 0 = 0.09375 + -0.125a + 0.03125a2 Solving for variable 'a'. Combine like terms: 0 + -0.09375 = -0.09375 -0.09375 + 0.125a + -0.03125a2 = 0.09375 + -0.125a + 0.03125a2 + -0.09375 + 0.125a + -0.03125a2 Reorder the terms: -0.09375 + 0.125a + -0.03125a2 = 0.09375 + -0.09375 + -0.125a + 0.125a + 0.03125a2 + -0.03125a2 Combine like terms: 0.09375 + -0.09375 = 0.00000 -0.09375 + 0.125a + -0.03125a2 = 0.00000 + -0.125a + 0.125a + 0.03125a2 + -0.03125a2 -0.09375 + 0.125a + -0.03125a2 = -0.125a + 0.125a + 0.03125a2 + -0.03125a2 Combine like terms: -0.125a + 0.125a = 0.000 -0.09375 + 0.125a + -0.03125a2 = 0.000 + 0.03125a2 + -0.03125a2 -0.09375 + 0.125a + -0.03125a2 = 0.03125a2 + -0.03125a2 Combine like terms: 0.03125a2 + -0.03125a2 = 0.00000 -0.09375 + 0.125a + -0.03125a2 = 0.00000 Begin completing the square. Divide all terms by -0.03125 the coefficient of the squared term: Divide each side by '-0.03125'. 3 + -4a + a2 = 0 Move the constant term to the right: Add '-3' to each side of the equation. 3 + -4a + -3 + a2 = 0 + -3 Reorder the terms: 3 + -3 + -4a + a2 = 0 + -3 Combine like terms: 3 + -3 = 0 0 + -4a + a2 = 0 + -3 -4a + a2 = 0 + -3 Combine like terms: 0 + -3 = -3 -4a + a2 = -3 The a term is -4a. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4a + 4 + a2 = -3 + 4 Reorder the terms: 4 + -4a + a2 = -3 + 4 Combine like terms: -3 + 4 = 1 4 + -4a + a2 = 1 Factor a perfect square on the left side: (a + -2)(a + -2) = 1 Calculate the square root of the right side: 1 Break this problem into two subproblems by setting (a + -2) equal to 1 and -1.Subproblem 1
a + -2 = 1 Simplifying a + -2 = 1 Reorder the terms: -2 + a = 1 Solving -2 + a = 1 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + a = 1 + 2 Combine like terms: -2 + 2 = 0 0 + a = 1 + 2 a = 1 + 2 Combine like terms: 1 + 2 = 3 a = 3 Simplifying a = 3Subproblem 2
a + -2 = -1 Simplifying a + -2 = -1 Reorder the terms: -2 + a = -1 Solving -2 + a = -1 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + a = -1 + 2 Combine like terms: -2 + 2 = 0 0 + a = -1 + 2 a = -1 + 2 Combine like terms: -1 + 2 = 1 a = 1 Simplifying a = 1Solution
The solution to the problem is based on the solutions from the subproblems. a = {3, 1}
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