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Simplifying (a + -10)(a + -14) = 1 Reorder the terms: (-10 + a)(a + -14) = 1 Reorder the terms: (-10 + a)(-14 + a) = 1 Multiply (-10 + a) * (-14 + a) (-10(-14 + a) + a(-14 + a)) = 1 ((-14 * -10 + a * -10) + a(-14 + a)) = 1 ((140 + -10a) + a(-14 + a)) = 1 (140 + -10a + (-14 * a + a * a)) = 1 (140 + -10a + (-14a + a2)) = 1 Combine like terms: -10a + -14a = -24a (140 + -24a + a2) = 1 Solving 140 + -24a + a2 = 1 Solving for variable 'a'. Reorder the terms: 140 + -1 + -24a + a2 = 1 + -1 Combine like terms: 140 + -1 = 139 139 + -24a + a2 = 1 + -1 Combine like terms: 1 + -1 = 0 139 + -24a + a2 = 0 Begin completing the square. Move the constant term to the right: Add '-139' to each side of the equation. 139 + -24a + -139 + a2 = 0 + -139 Reorder the terms: 139 + -139 + -24a + a2 = 0 + -139 Combine like terms: 139 + -139 = 0 0 + -24a + a2 = 0 + -139 -24a + a2 = 0 + -139 Combine like terms: 0 + -139 = -139 -24a + a2 = -139 The a term is -24a. Take half its coefficient (-12). Square it (144) and add it to both sides. Add '144' to each side of the equation. -24a + 144 + a2 = -139 + 144 Reorder the terms: 144 + -24a + a2 = -139 + 144 Combine like terms: -139 + 144 = 5 144 + -24a + a2 = 5 Factor a perfect square on the left side: (a + -12)(a + -12) = 5 Calculate the square root of the right side: 2.236067978 Break this problem into two subproblems by setting (a + -12) equal to 2.236067978 and -2.236067978.Subproblem 1
a + -12 = 2.236067978 Simplifying a + -12 = 2.236067978 Reorder the terms: -12 + a = 2.236067978 Solving -12 + a = 2.236067978 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '12' to each side of the equation. -12 + 12 + a = 2.236067978 + 12 Combine like terms: -12 + 12 = 0 0 + a = 2.236067978 + 12 a = 2.236067978 + 12 Combine like terms: 2.236067978 + 12 = 14.236067978 a = 14.236067978 Simplifying a = 14.236067978Subproblem 2
a + -12 = -2.236067978 Simplifying a + -12 = -2.236067978 Reorder the terms: -12 + a = -2.236067978 Solving -12 + a = -2.236067978 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '12' to each side of the equation. -12 + 12 + a = -2.236067978 + 12 Combine like terms: -12 + 12 = 0 0 + a = -2.236067978 + 12 a = -2.236067978 + 12 Combine like terms: -2.236067978 + 12 = 9.763932022 a = 9.763932022 Simplifying a = 9.763932022Solution
The solution to the problem is based on the solutions from the subproblems. a = {14.236067978, 9.763932022}
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