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Simplifying (0.7a + -0.21)(0.05a + 1) = 0 Reorder the terms: (-0.21 + 0.7a)(0.05a + 1) = 0 Reorder the terms: (-0.21 + 0.7a)(1 + 0.05a) = 0 Multiply (-0.21 + 0.7a) * (1 + 0.05a) (-0.21(1 + 0.05a) + 0.7a * (1 + 0.05a)) = 0 ((1 * -0.21 + 0.05a * -0.21) + 0.7a * (1 + 0.05a)) = 0 ((-0.21 + -0.0105a) + 0.7a * (1 + 0.05a)) = 0 (-0.21 + -0.0105a + (1 * 0.7a + 0.05a * 0.7a)) = 0 (-0.21 + -0.0105a + (0.7a + 0.035a2)) = 0 Combine like terms: -0.0105a + 0.7a = 0.6895a (-0.21 + 0.6895a + 0.035a2) = 0 Solving -0.21 + 0.6895a + 0.035a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 0.035 the coefficient of the squared term: Divide each side by '0.035'. -6 + 19.7a + a2 = 0 Move the constant term to the right: Add '6' to each side of the equation. -6 + 19.7a + 6 + a2 = 0 + 6 Reorder the terms: -6 + 6 + 19.7a + a2 = 0 + 6 Combine like terms: -6 + 6 = 0 0 + 19.7a + a2 = 0 + 6 19.7a + a2 = 0 + 6 Combine like terms: 0 + 6 = 6 19.7a + a2 = 6 The a term is 19.7a. Take half its coefficient (9.85). Square it (97.0225) and add it to both sides. Add '97.0225' to each side of the equation. 19.7a + 97.0225 + a2 = 6 + 97.0225 Reorder the terms: 97.0225 + 19.7a + a2 = 6 + 97.0225 Combine like terms: 6 + 97.0225 = 103.0225 97.0225 + 19.7a + a2 = 103.0225 Factor a perfect square on the left side: (a + 9.85)(a + 9.85) = 103.0225 Calculate the square root of the right side: 10.15 Break this problem into two subproblems by setting (a + 9.85) equal to 10.15 and -10.15.Subproblem 1
a + 9.85 = 10.15 Simplifying a + 9.85 = 10.15 Reorder the terms: 9.85 + a = 10.15 Solving 9.85 + a = 10.15 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-9.85' to each side of the equation. 9.85 + -9.85 + a = 10.15 + -9.85 Combine like terms: 9.85 + -9.85 = 0.00 0.00 + a = 10.15 + -9.85 a = 10.15 + -9.85 Combine like terms: 10.15 + -9.85 = 0.3 a = 0.3 Simplifying a = 0.3Subproblem 2
a + 9.85 = -10.15 Simplifying a + 9.85 = -10.15 Reorder the terms: 9.85 + a = -10.15 Solving 9.85 + a = -10.15 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-9.85' to each side of the equation. 9.85 + -9.85 + a = -10.15 + -9.85 Combine like terms: 9.85 + -9.85 = 0.00 0.00 + a = -10.15 + -9.85 a = -10.15 + -9.85 Combine like terms: -10.15 + -9.85 = -20 a = -20 Simplifying a = -20Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.3, -20}
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