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Simplifying 13.6 = (4 + -2a)(10 + -2a) Multiply (4 + -2a) * (10 + -2a) 13.6 = (4(10 + -2a) + -2a * (10 + -2a)) 13.6 = ((10 * 4 + -2a * 4) + -2a * (10 + -2a)) 13.6 = ((40 + -8a) + -2a * (10 + -2a)) 13.6 = (40 + -8a + (10 * -2a + -2a * -2a)) 13.6 = (40 + -8a + (-20a + 4a2)) Combine like terms: -8a + -20a = -28a 13.6 = (40 + -28a + 4a2) Solving 13.6 = 40 + -28a + 4a2 Solving for variable 'a'. Combine like terms: 13.6 + -40 = -26.4 -26.4 + 28a + -4a2 = 40 + -28a + 4a2 + -40 + 28a + -4a2 Reorder the terms: -26.4 + 28a + -4a2 = 40 + -40 + -28a + 28a + 4a2 + -4a2 Combine like terms: 40 + -40 = 0 -26.4 + 28a + -4a2 = 0 + -28a + 28a + 4a2 + -4a2 -26.4 + 28a + -4a2 = -28a + 28a + 4a2 + -4a2 Combine like terms: -28a + 28a = 0 -26.4 + 28a + -4a2 = 0 + 4a2 + -4a2 -26.4 + 28a + -4a2 = 4a2 + -4a2 Combine like terms: 4a2 + -4a2 = 0 -26.4 + 28a + -4a2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. 6.6 + -7a + a2 = 0 Move the constant term to the right: Add '-6.6' to each side of the equation. 6.6 + -7a + -6.6 + a2 = 0 + -6.6 Reorder the terms: 6.6 + -6.6 + -7a + a2 = 0 + -6.6 Combine like terms: 6.6 + -6.6 = 0.0 0.0 + -7a + a2 = 0 + -6.6 -7a + a2 = 0 + -6.6 Combine like terms: 0 + -6.6 = -6.6 -7a + a2 = -6.6 The a term is -7a. Take half its coefficient (-3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. -7a + 12.25 + a2 = -6.6 + 12.25 Reorder the terms: 12.25 + -7a + a2 = -6.6 + 12.25 Combine like terms: -6.6 + 12.25 = 5.65 12.25 + -7a + a2 = 5.65 Factor a perfect square on the left side: (a + -3.5)(a + -3.5) = 5.65 Calculate the square root of the right side: 2.376972865 Break this problem into two subproblems by setting (a + -3.5) equal to 2.376972865 and -2.376972865.Subproblem 1
a + -3.5 = 2.376972865 Simplifying a + -3.5 = 2.376972865 Reorder the terms: -3.5 + a = 2.376972865 Solving -3.5 + a = 2.376972865 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + a = 2.376972865 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + a = 2.376972865 + 3.5 a = 2.376972865 + 3.5 Combine like terms: 2.376972865 + 3.5 = 5.876972865 a = 5.876972865 Simplifying a = 5.876972865Subproblem 2
a + -3.5 = -2.376972865 Simplifying a + -3.5 = -2.376972865 Reorder the terms: -3.5 + a = -2.376972865 Solving -3.5 + a = -2.376972865 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + a = -2.376972865 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + a = -2.376972865 + 3.5 a = -2.376972865 + 3.5 Combine like terms: -2.376972865 + 3.5 = 1.123027135 a = 1.123027135 Simplifying a = 1.123027135Solution
The solution to the problem is based on the solutions from the subproblems. a = {5.876972865, 1.123027135}
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