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Simplifying (0.2a + 1.3)(0.5a + -0.1) = 0 Reorder the terms: (1.3 + 0.2a)(0.5a + -0.1) = 0 Reorder the terms: (1.3 + 0.2a)(-0.1 + 0.5a) = 0 Multiply (1.3 + 0.2a) * (-0.1 + 0.5a) (1.3(-0.1 + 0.5a) + 0.2a * (-0.1 + 0.5a)) = 0 ((-0.1 * 1.3 + 0.5a * 1.3) + 0.2a * (-0.1 + 0.5a)) = 0 ((-0.13 + 0.65a) + 0.2a * (-0.1 + 0.5a)) = 0 (-0.13 + 0.65a + (-0.1 * 0.2a + 0.5a * 0.2a)) = 0 (-0.13 + 0.65a + (-0.02a + 0.1a2)) = 0 Combine like terms: 0.65a + -0.02a = 0.63a (-0.13 + 0.63a + 0.1a2) = 0 Solving -0.13 + 0.63a + 0.1a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 0.1 the coefficient of the squared term: Divide each side by '0.1'. -1.3 + 6.3a + a2 = 0 Move the constant term to the right: Add '1.3' to each side of the equation. -1.3 + 6.3a + 1.3 + a2 = 0 + 1.3 Reorder the terms: -1.3 + 1.3 + 6.3a + a2 = 0 + 1.3 Combine like terms: -1.3 + 1.3 = 0.0 0.0 + 6.3a + a2 = 0 + 1.3 6.3a + a2 = 0 + 1.3 Combine like terms: 0 + 1.3 = 1.3 6.3a + a2 = 1.3 The a term is 6.3a. Take half its coefficient (3.15). Square it (9.9225) and add it to both sides. Add '9.9225' to each side of the equation. 6.3a + 9.9225 + a2 = 1.3 + 9.9225 Reorder the terms: 9.9225 + 6.3a + a2 = 1.3 + 9.9225 Combine like terms: 1.3 + 9.9225 = 11.2225 9.9225 + 6.3a + a2 = 11.2225 Factor a perfect square on the left side: (a + 3.15)(a + 3.15) = 11.2225 Calculate the square root of the right side: 3.35 Break this problem into two subproblems by setting (a + 3.15) equal to 3.35 and -3.35.Subproblem 1
a + 3.15 = 3.35 Simplifying a + 3.15 = 3.35 Reorder the terms: 3.15 + a = 3.35 Solving 3.15 + a = 3.35 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-3.15' to each side of the equation. 3.15 + -3.15 + a = 3.35 + -3.15 Combine like terms: 3.15 + -3.15 = 0.00 0.00 + a = 3.35 + -3.15 a = 3.35 + -3.15 Combine like terms: 3.35 + -3.15 = 0.2 a = 0.2 Simplifying a = 0.2Subproblem 2
a + 3.15 = -3.35 Simplifying a + 3.15 = -3.35 Reorder the terms: 3.15 + a = -3.35 Solving 3.15 + a = -3.35 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-3.15' to each side of the equation. 3.15 + -3.15 + a = -3.35 + -3.15 Combine like terms: 3.15 + -3.15 = 0.00 0.00 + a = -3.35 + -3.15 a = -3.35 + -3.15 Combine like terms: -3.35 + -3.15 = -6.5 a = -6.5 Simplifying a = -6.5Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.2, -6.5}
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