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Simplifying [5a2 + -20a + 20] + [3a + 3] = 25 Reorder the terms: [20 + -20a + 5a2] + [3a + 3] = 25 Remove brackets around [20 + -20a + 5a2] 20 + -20a + 5a2 + [3a + 3] = 25 Reorder the terms: 20 + -20a + 5a2 + [3 + 3a] = 25 Remove brackets around [3 + 3a] 20 + -20a + 5a2 + 3 + 3a = 25 Reorder the terms: 20 + 3 + -20a + 3a + 5a2 = 25 Combine like terms: 20 + 3 = 23 23 + -20a + 3a + 5a2 = 25 Combine like terms: -20a + 3a = -17a 23 + -17a + 5a2 = 25 Solving 23 + -17a + 5a2 = 25 Solving for variable 'a'. Reorder the terms: 23 + -25 + -17a + 5a2 = 25 + -25 Combine like terms: 23 + -25 = -2 -2 + -17a + 5a2 = 25 + -25 Combine like terms: 25 + -25 = 0 -2 + -17a + 5a2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -0.4 + -3.4a + a2 = 0 Move the constant term to the right: Add '0.4' to each side of the equation. -0.4 + -3.4a + 0.4 + a2 = 0 + 0.4 Reorder the terms: -0.4 + 0.4 + -3.4a + a2 = 0 + 0.4 Combine like terms: -0.4 + 0.4 = 0.0 0.0 + -3.4a + a2 = 0 + 0.4 -3.4a + a2 = 0 + 0.4 Combine like terms: 0 + 0.4 = 0.4 -3.4a + a2 = 0.4 The a term is -3.4a. Take half its coefficient (-1.7). Square it (2.89) and add it to both sides. Add '2.89' to each side of the equation. -3.4a + 2.89 + a2 = 0.4 + 2.89 Reorder the terms: 2.89 + -3.4a + a2 = 0.4 + 2.89 Combine like terms: 0.4 + 2.89 = 3.29 2.89 + -3.4a + a2 = 3.29 Factor a perfect square on the left side: (a + -1.7)(a + -1.7) = 3.29 Calculate the square root of the right side: 1.813835715 Break this problem into two subproblems by setting (a + -1.7) equal to 1.813835715 and -1.813835715.Subproblem 1
a + -1.7 = 1.813835715 Simplifying a + -1.7 = 1.813835715 Reorder the terms: -1.7 + a = 1.813835715 Solving -1.7 + a = 1.813835715 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.7' to each side of the equation. -1.7 + 1.7 + a = 1.813835715 + 1.7 Combine like terms: -1.7 + 1.7 = 0.0 0.0 + a = 1.813835715 + 1.7 a = 1.813835715 + 1.7 Combine like terms: 1.813835715 + 1.7 = 3.513835715 a = 3.513835715 Simplifying a = 3.513835715Subproblem 2
a + -1.7 = -1.813835715 Simplifying a + -1.7 = -1.813835715 Reorder the terms: -1.7 + a = -1.813835715 Solving -1.7 + a = -1.813835715 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.7' to each side of the equation. -1.7 + 1.7 + a = -1.813835715 + 1.7 Combine like terms: -1.7 + 1.7 = 0.0 0.0 + a = -1.813835715 + 1.7 a = -1.813835715 + 1.7 Combine like terms: -1.813835715 + 1.7 = -0.113835715 a = -0.113835715 Simplifying a = -0.113835715Solution
The solution to the problem is based on the solutions from the subproblems. a = {3.513835715, -0.113835715}
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