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Simplifying 10 = (a + -2)(2a + 1) Reorder the terms: 10 = (-2 + a)(2a + 1) Reorder the terms: 10 = (-2 + a)(1 + 2a) Multiply (-2 + a) * (1 + 2a) 10 = (-2(1 + 2a) + a(1 + 2a)) 10 = ((1 * -2 + 2a * -2) + a(1 + 2a)) 10 = ((-2 + -4a) + a(1 + 2a)) 10 = (-2 + -4a + (1 * a + 2a * a)) 10 = (-2 + -4a + (1a + 2a2)) Combine like terms: -4a + 1a = -3a 10 = (-2 + -3a + 2a2) Solving 10 = -2 + -3a + 2a2 Solving for variable 'a'. Combine like terms: 10 + 2 = 12 12 + 3a + -2a2 = -2 + -3a + 2a2 + 2 + 3a + -2a2 Reorder the terms: 12 + 3a + -2a2 = -2 + 2 + -3a + 3a + 2a2 + -2a2 Combine like terms: -2 + 2 = 0 12 + 3a + -2a2 = 0 + -3a + 3a + 2a2 + -2a2 12 + 3a + -2a2 = -3a + 3a + 2a2 + -2a2 Combine like terms: -3a + 3a = 0 12 + 3a + -2a2 = 0 + 2a2 + -2a2 12 + 3a + -2a2 = 2a2 + -2a2 Combine like terms: 2a2 + -2a2 = 0 12 + 3a + -2a2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -6 + -1.5a + a2 = 0 Move the constant term to the right: Add '6' to each side of the equation. -6 + -1.5a + 6 + a2 = 0 + 6 Reorder the terms: -6 + 6 + -1.5a + a2 = 0 + 6 Combine like terms: -6 + 6 = 0 0 + -1.5a + a2 = 0 + 6 -1.5a + a2 = 0 + 6 Combine like terms: 0 + 6 = 6 -1.5a + a2 = 6 The a term is -1.5a. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5a + 0.5625 + a2 = 6 + 0.5625 Reorder the terms: 0.5625 + -1.5a + a2 = 6 + 0.5625 Combine like terms: 6 + 0.5625 = 6.5625 0.5625 + -1.5a + a2 = 6.5625 Factor a perfect square on the left side: (a + -0.75)(a + -0.75) = 6.5625 Calculate the square root of the right side: 2.561737691 Break this problem into two subproblems by setting (a + -0.75) equal to 2.561737691 and -2.561737691.Subproblem 1
a + -0.75 = 2.561737691 Simplifying a + -0.75 = 2.561737691 Reorder the terms: -0.75 + a = 2.561737691 Solving -0.75 + a = 2.561737691 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + a = 2.561737691 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + a = 2.561737691 + 0.75 a = 2.561737691 + 0.75 Combine like terms: 2.561737691 + 0.75 = 3.311737691 a = 3.311737691 Simplifying a = 3.311737691Subproblem 2
a + -0.75 = -2.561737691 Simplifying a + -0.75 = -2.561737691 Reorder the terms: -0.75 + a = -2.561737691 Solving -0.75 + a = -2.561737691 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + a = -2.561737691 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + a = -2.561737691 + 0.75 a = -2.561737691 + 0.75 Combine like terms: -2.561737691 + 0.75 = -1.811737691 a = -1.811737691 Simplifying a = -1.811737691Solution
The solution to the problem is based on the solutions from the subproblems. a = {3.311737691, -1.811737691}
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