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Simplifying 7(a + 1) + -2a = 5 + 4a(a + -1) Reorder the terms: 7(1 + a) + -2a = 5 + 4a(a + -1) (1 * 7 + a * 7) + -2a = 5 + 4a(a + -1) (7 + 7a) + -2a = 5 + 4a(a + -1) Combine like terms: 7a + -2a = 5a 7 + 5a = 5 + 4a(a + -1) Reorder the terms: 7 + 5a = 5 + 4a(-1 + a) 7 + 5a = 5 + (-1 * 4a + a * 4a) 7 + 5a = 5 + (-4a + 4a2) Solving 7 + 5a = 5 + -4a + 4a2 Solving for variable 'a'. Reorder the terms: 7 + -5 + 5a + 4a + -4a2 = 5 + -4a + 4a2 + -5 + 4a + -4a2 Combine like terms: 7 + -5 = 2 2 + 5a + 4a + -4a2 = 5 + -4a + 4a2 + -5 + 4a + -4a2 Combine like terms: 5a + 4a = 9a 2 + 9a + -4a2 = 5 + -4a + 4a2 + -5 + 4a + -4a2 Reorder the terms: 2 + 9a + -4a2 = 5 + -5 + -4a + 4a + 4a2 + -4a2 Combine like terms: 5 + -5 = 0 2 + 9a + -4a2 = 0 + -4a + 4a + 4a2 + -4a2 2 + 9a + -4a2 = -4a + 4a + 4a2 + -4a2 Combine like terms: -4a + 4a = 0 2 + 9a + -4a2 = 0 + 4a2 + -4a2 2 + 9a + -4a2 = 4a2 + -4a2 Combine like terms: 4a2 + -4a2 = 0 2 + 9a + -4a2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -0.5 + -2.25a + a2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + -2.25a + 0.5 + a2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + -2.25a + a2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + -2.25a + a2 = 0 + 0.5 -2.25a + a2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 -2.25a + a2 = 0.5 The a term is -2.25a. Take half its coefficient (-1.125). Square it (1.265625) and add it to both sides. Add '1.265625' to each side of the equation. -2.25a + 1.265625 + a2 = 0.5 + 1.265625 Reorder the terms: 1.265625 + -2.25a + a2 = 0.5 + 1.265625 Combine like terms: 0.5 + 1.265625 = 1.765625 1.265625 + -2.25a + a2 = 1.765625 Factor a perfect square on the left side: (a + -1.125)(a + -1.125) = 1.765625 Calculate the square root of the right side: 1.328768227 Break this problem into two subproblems by setting (a + -1.125) equal to 1.328768227 and -1.328768227.Subproblem 1
a + -1.125 = 1.328768227 Simplifying a + -1.125 = 1.328768227 Reorder the terms: -1.125 + a = 1.328768227 Solving -1.125 + a = 1.328768227 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.125' to each side of the equation. -1.125 + 1.125 + a = 1.328768227 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + a = 1.328768227 + 1.125 a = 1.328768227 + 1.125 Combine like terms: 1.328768227 + 1.125 = 2.453768227 a = 2.453768227 Simplifying a = 2.453768227Subproblem 2
a + -1.125 = -1.328768227 Simplifying a + -1.125 = -1.328768227 Reorder the terms: -1.125 + a = -1.328768227 Solving -1.125 + a = -1.328768227 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.125' to each side of the equation. -1.125 + 1.125 + a = -1.328768227 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + a = -1.328768227 + 1.125 a = -1.328768227 + 1.125 Combine like terms: -1.328768227 + 1.125 = -0.203768227 a = -0.203768227 Simplifying a = -0.203768227Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.453768227, -0.203768227}
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