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Simplifying 30 + 45a(4a + -80) = -35(60a + 70) + -40a Reorder the terms: 30 + 45a(-80 + 4a) = -35(60a + 70) + -40a 30 + (-80 * 45a + 4a * 45a) = -35(60a + 70) + -40a 30 + (-3600a + 180a2) = -35(60a + 70) + -40a Reorder the terms: 30 + -3600a + 180a2 = -35(70 + 60a) + -40a 30 + -3600a + 180a2 = (70 * -35 + 60a * -35) + -40a 30 + -3600a + 180a2 = (-2450 + -2100a) + -40a Combine like terms: -2100a + -40a = -2140a 30 + -3600a + 180a2 = -2450 + -2140a Solving 30 + -3600a + 180a2 = -2450 + -2140a Solving for variable 'a'. Reorder the terms: 30 + 2450 + -3600a + 2140a + 180a2 = -2450 + -2140a + 2450 + 2140a Combine like terms: 30 + 2450 = 2480 2480 + -3600a + 2140a + 180a2 = -2450 + -2140a + 2450 + 2140a Combine like terms: -3600a + 2140a = -1460a 2480 + -1460a + 180a2 = -2450 + -2140a + 2450 + 2140a Reorder the terms: 2480 + -1460a + 180a2 = -2450 + 2450 + -2140a + 2140a Combine like terms: -2450 + 2450 = 0 2480 + -1460a + 180a2 = 0 + -2140a + 2140a 2480 + -1460a + 180a2 = -2140a + 2140a Combine like terms: -2140a + 2140a = 0 2480 + -1460a + 180a2 = 0 Factor out the Greatest Common Factor (GCF), '20'. 20(124 + -73a + 9a2) = 0 Ignore the factor 20.Subproblem 1
Set the factor '(124 + -73a + 9a2)' equal to zero and attempt to solve: Simplifying 124 + -73a + 9a2 = 0 Solving 124 + -73a + 9a2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. 13.77777778 + -8.111111111a + a2 = 0 Move the constant term to the right: Add '-13.77777778' to each side of the equation. 13.77777778 + -8.111111111a + -13.77777778 + a2 = 0 + -13.77777778 Reorder the terms: 13.77777778 + -13.77777778 + -8.111111111a + a2 = 0 + -13.77777778 Combine like terms: 13.77777778 + -13.77777778 = 0.00000000 0.00000000 + -8.111111111a + a2 = 0 + -13.77777778 -8.111111111a + a2 = 0 + -13.77777778 Combine like terms: 0 + -13.77777778 = -13.77777778 -8.111111111a + a2 = -13.77777778 The a term is -8.111111111a. Take half its coefficient (-4.055555556). Square it (16.44753087) and add it to both sides. Add '16.44753087' to each side of the equation. -8.111111111a + 16.44753087 + a2 = -13.77777778 + 16.44753087 Reorder the terms: 16.44753087 + -8.111111111a + a2 = -13.77777778 + 16.44753087 Combine like terms: -13.77777778 + 16.44753087 = 2.66975309 16.44753087 + -8.111111111a + a2 = 2.66975309 Factor a perfect square on the left side: (a + -4.055555556)(a + -4.055555556) = 2.66975309 Calculate the square root of the right side: 1.633937909 Break this problem into two subproblems by setting (a + -4.055555556) equal to 1.633937909 and -1.633937909.Subproblem 1
a + -4.055555556 = 1.633937909 Simplifying a + -4.055555556 = 1.633937909 Reorder the terms: -4.055555556 + a = 1.633937909 Solving -4.055555556 + a = 1.633937909 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '4.055555556' to each side of the equation. -4.055555556 + 4.055555556 + a = 1.633937909 + 4.055555556 Combine like terms: -4.055555556 + 4.055555556 = 0.000000000 0.000000000 + a = 1.633937909 + 4.055555556 a = 1.633937909 + 4.055555556 Combine like terms: 1.633937909 + 4.055555556 = 5.689493465 a = 5.689493465 Simplifying a = 5.689493465Subproblem 2
a + -4.055555556 = -1.633937909 Simplifying a + -4.055555556 = -1.633937909 Reorder the terms: -4.055555556 + a = -1.633937909 Solving -4.055555556 + a = -1.633937909 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '4.055555556' to each side of the equation. -4.055555556 + 4.055555556 + a = -1.633937909 + 4.055555556 Combine like terms: -4.055555556 + 4.055555556 = 0.000000000 0.000000000 + a = -1.633937909 + 4.055555556 a = -1.633937909 + 4.055555556 Combine like terms: -1.633937909 + 4.055555556 = 2.421617647 a = 2.421617647 Simplifying a = 2.421617647Solution
The solution to the problem is based on the solutions from the subproblems. a = {5.689493465, 2.421617647}Solution
a = {5.689493465, 2.421617647}
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