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Simplifying 2a(a + -2) = 3a + 13 Reorder the terms: 2a(-2 + a) = 3a + 13 (-2 * 2a + a * 2a) = 3a + 13 (-4a + 2a2) = 3a + 13 Reorder the terms: -4a + 2a2 = 13 + 3a Solving -4a + 2a2 = 13 + 3a Solving for variable 'a'. Reorder the terms: -13 + -4a + -3a + 2a2 = 13 + 3a + -13 + -3a Combine like terms: -4a + -3a = -7a -13 + -7a + 2a2 = 13 + 3a + -13 + -3a Reorder the terms: -13 + -7a + 2a2 = 13 + -13 + 3a + -3a Combine like terms: 13 + -13 = 0 -13 + -7a + 2a2 = 0 + 3a + -3a -13 + -7a + 2a2 = 3a + -3a Combine like terms: 3a + -3a = 0 -13 + -7a + 2a2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -6.5 + -3.5a + a2 = 0 Move the constant term to the right: Add '6.5' to each side of the equation. -6.5 + -3.5a + 6.5 + a2 = 0 + 6.5 Reorder the terms: -6.5 + 6.5 + -3.5a + a2 = 0 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + -3.5a + a2 = 0 + 6.5 -3.5a + a2 = 0 + 6.5 Combine like terms: 0 + 6.5 = 6.5 -3.5a + a2 = 6.5 The a term is -3.5a. Take half its coefficient (-1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. -3.5a + 3.0625 + a2 = 6.5 + 3.0625 Reorder the terms: 3.0625 + -3.5a + a2 = 6.5 + 3.0625 Combine like terms: 6.5 + 3.0625 = 9.5625 3.0625 + -3.5a + a2 = 9.5625 Factor a perfect square on the left side: (a + -1.75)(a + -1.75) = 9.5625 Calculate the square root of the right side: 3.092329219 Break this problem into two subproblems by setting (a + -1.75) equal to 3.092329219 and -3.092329219.Subproblem 1
a + -1.75 = 3.092329219 Simplifying a + -1.75 = 3.092329219 Reorder the terms: -1.75 + a = 3.092329219 Solving -1.75 + a = 3.092329219 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + a = 3.092329219 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + a = 3.092329219 + 1.75 a = 3.092329219 + 1.75 Combine like terms: 3.092329219 + 1.75 = 4.842329219 a = 4.842329219 Simplifying a = 4.842329219Subproblem 2
a + -1.75 = -3.092329219 Simplifying a + -1.75 = -3.092329219 Reorder the terms: -1.75 + a = -3.092329219 Solving -1.75 + a = -3.092329219 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + a = -3.092329219 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + a = -3.092329219 + 1.75 a = -3.092329219 + 1.75 Combine like terms: -3.092329219 + 1.75 = -1.342329219 a = -1.342329219 Simplifying a = -1.342329219Solution
The solution to the problem is based on the solutions from the subproblems. a = {4.842329219, -1.342329219}
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