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Simplifying 7a + 29 = a2 + 6a + 7 Reorder the terms: 29 + 7a = a2 + 6a + 7 Reorder the terms: 29 + 7a = 7 + 6a + a2 Solving 29 + 7a = 7 + 6a + a2 Solving for variable 'a'. Reorder the terms: 29 + -7 + 7a + -6a + -1a2 = 7 + 6a + a2 + -7 + -6a + -1a2 Combine like terms: 29 + -7 = 22 22 + 7a + -6a + -1a2 = 7 + 6a + a2 + -7 + -6a + -1a2 Combine like terms: 7a + -6a = 1a 22 + 1a + -1a2 = 7 + 6a + a2 + -7 + -6a + -1a2 Reorder the terms: 22 + 1a + -1a2 = 7 + -7 + 6a + -6a + a2 + -1a2 Combine like terms: 7 + -7 = 0 22 + 1a + -1a2 = 0 + 6a + -6a + a2 + -1a2 22 + 1a + -1a2 = 6a + -6a + a2 + -1a2 Combine like terms: 6a + -6a = 0 22 + 1a + -1a2 = 0 + a2 + -1a2 22 + 1a + -1a2 = a2 + -1a2 Combine like terms: a2 + -1a2 = 0 22 + 1a + -1a2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -22 + -1a + a2 = 0 Move the constant term to the right: Add '22' to each side of the equation. -22 + -1a + 22 + a2 = 0 + 22 Reorder the terms: -22 + 22 + -1a + a2 = 0 + 22 Combine like terms: -22 + 22 = 0 0 + -1a + a2 = 0 + 22 -1a + a2 = 0 + 22 Combine like terms: 0 + 22 = 22 -1a + a2 = 22 The a term is -1a. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1a + 0.25 + a2 = 22 + 0.25 Reorder the terms: 0.25 + -1a + a2 = 22 + 0.25 Combine like terms: 22 + 0.25 = 22.25 0.25 + -1a + a2 = 22.25 Factor a perfect square on the left side: (a + -0.5)(a + -0.5) = 22.25 Calculate the square root of the right side: 4.716990566 Break this problem into two subproblems by setting (a + -0.5) equal to 4.716990566 and -4.716990566.Subproblem 1
a + -0.5 = 4.716990566 Simplifying a + -0.5 = 4.716990566 Reorder the terms: -0.5 + a = 4.716990566 Solving -0.5 + a = 4.716990566 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + a = 4.716990566 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + a = 4.716990566 + 0.5 a = 4.716990566 + 0.5 Combine like terms: 4.716990566 + 0.5 = 5.216990566 a = 5.216990566 Simplifying a = 5.216990566Subproblem 2
a + -0.5 = -4.716990566 Simplifying a + -0.5 = -4.716990566 Reorder the terms: -0.5 + a = -4.716990566 Solving -0.5 + a = -4.716990566 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + a = -4.716990566 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + a = -4.716990566 + 0.5 a = -4.716990566 + 0.5 Combine like terms: -4.716990566 + 0.5 = -4.216990566 a = -4.216990566 Simplifying a = -4.216990566Solution
The solution to the problem is based on the solutions from the subproblems. a = {5.216990566, -4.216990566}
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