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Simplifying (a + 5)(a + -2) = 28 Reorder the terms: (5 + a)(a + -2) = 28 Reorder the terms: (5 + a)(-2 + a) = 28 Multiply (5 + a) * (-2 + a) (5(-2 + a) + a(-2 + a)) = 28 ((-2 * 5 + a * 5) + a(-2 + a)) = 28 ((-10 + 5a) + a(-2 + a)) = 28 (-10 + 5a + (-2 * a + a * a)) = 28 (-10 + 5a + (-2a + a2)) = 28 Combine like terms: 5a + -2a = 3a (-10 + 3a + a2) = 28 Solving -10 + 3a + a2 = 28 Solving for variable 'a'. Reorder the terms: -10 + -28 + 3a + a2 = 28 + -28 Combine like terms: -10 + -28 = -38 -38 + 3a + a2 = 28 + -28 Combine like terms: 28 + -28 = 0 -38 + 3a + a2 = 0 Begin completing the square. Move the constant term to the right: Add '38' to each side of the equation. -38 + 3a + 38 + a2 = 0 + 38 Reorder the terms: -38 + 38 + 3a + a2 = 0 + 38 Combine like terms: -38 + 38 = 0 0 + 3a + a2 = 0 + 38 3a + a2 = 0 + 38 Combine like terms: 0 + 38 = 38 3a + a2 = 38 The a term is 3a. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3a + 2.25 + a2 = 38 + 2.25 Reorder the terms: 2.25 + 3a + a2 = 38 + 2.25 Combine like terms: 38 + 2.25 = 40.25 2.25 + 3a + a2 = 40.25 Factor a perfect square on the left side: (a + 1.5)(a + 1.5) = 40.25 Calculate the square root of the right side: 6.34428877 Break this problem into two subproblems by setting (a + 1.5) equal to 6.34428877 and -6.34428877.Subproblem 1
a + 1.5 = 6.34428877 Simplifying a + 1.5 = 6.34428877 Reorder the terms: 1.5 + a = 6.34428877 Solving 1.5 + a = 6.34428877 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + a = 6.34428877 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + a = 6.34428877 + -1.5 a = 6.34428877 + -1.5 Combine like terms: 6.34428877 + -1.5 = 4.84428877 a = 4.84428877 Simplifying a = 4.84428877Subproblem 2
a + 1.5 = -6.34428877 Simplifying a + 1.5 = -6.34428877 Reorder the terms: 1.5 + a = -6.34428877 Solving 1.5 + a = -6.34428877 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + a = -6.34428877 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + a = -6.34428877 + -1.5 a = -6.34428877 + -1.5 Combine like terms: -6.34428877 + -1.5 = -7.84428877 a = -7.84428877 Simplifying a = -7.84428877Solution
The solution to the problem is based on the solutions from the subproblems. a = {4.84428877, -7.84428877}
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