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Simplifying 4(a + 5) + -5(a2 + -4a + 7) = 0 Reorder the terms: 4(5 + a) + -5(a2 + -4a + 7) = 0 (5 * 4 + a * 4) + -5(a2 + -4a + 7) = 0 (20 + 4a) + -5(a2 + -4a + 7) = 0 Reorder the terms: 20 + 4a + -5(7 + -4a + a2) = 0 20 + 4a + (7 * -5 + -4a * -5 + a2 * -5) = 0 20 + 4a + (-35 + 20a + -5a2) = 0 Reorder the terms: 20 + -35 + 4a + 20a + -5a2 = 0 Combine like terms: 20 + -35 = -15 -15 + 4a + 20a + -5a2 = 0 Combine like terms: 4a + 20a = 24a -15 + 24a + -5a2 = 0 Solving -15 + 24a + -5a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. 3 + -4.8a + a2 = 0 Move the constant term to the right: Add '-3' to each side of the equation. 3 + -4.8a + -3 + a2 = 0 + -3 Reorder the terms: 3 + -3 + -4.8a + a2 = 0 + -3 Combine like terms: 3 + -3 = 0 0 + -4.8a + a2 = 0 + -3 -4.8a + a2 = 0 + -3 Combine like terms: 0 + -3 = -3 -4.8a + a2 = -3 The a term is -4.8a. Take half its coefficient (-2.4). Square it (5.76) and add it to both sides. Add '5.76' to each side of the equation. -4.8a + 5.76 + a2 = -3 + 5.76 Reorder the terms: 5.76 + -4.8a + a2 = -3 + 5.76 Combine like terms: -3 + 5.76 = 2.76 5.76 + -4.8a + a2 = 2.76 Factor a perfect square on the left side: (a + -2.4)(a + -2.4) = 2.76 Calculate the square root of the right side: 1.661324773 Break this problem into two subproblems by setting (a + -2.4) equal to 1.661324773 and -1.661324773.Subproblem 1
a + -2.4 = 1.661324773 Simplifying a + -2.4 = 1.661324773 Reorder the terms: -2.4 + a = 1.661324773 Solving -2.4 + a = 1.661324773 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2.4' to each side of the equation. -2.4 + 2.4 + a = 1.661324773 + 2.4 Combine like terms: -2.4 + 2.4 = 0.0 0.0 + a = 1.661324773 + 2.4 a = 1.661324773 + 2.4 Combine like terms: 1.661324773 + 2.4 = 4.061324773 a = 4.061324773 Simplifying a = 4.061324773Subproblem 2
a + -2.4 = -1.661324773 Simplifying a + -2.4 = -1.661324773 Reorder the terms: -2.4 + a = -1.661324773 Solving -2.4 + a = -1.661324773 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2.4' to each side of the equation. -2.4 + 2.4 + a = -1.661324773 + 2.4 Combine like terms: -2.4 + 2.4 = 0.0 0.0 + a = -1.661324773 + 2.4 a = -1.661324773 + 2.4 Combine like terms: -1.661324773 + 2.4 = 0.738675227 a = 0.738675227 Simplifying a = 0.738675227Solution
The solution to the problem is based on the solutions from the subproblems. a = {4.061324773, 0.738675227}
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