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Simplifying -2a2 + 18a = 30 Reorder the terms: 18a + -2a2 = 30 Solving 18a + -2a2 = 30 Solving for variable 'a'. Reorder the terms: -30 + 18a + -2a2 = 30 + -30 Combine like terms: 30 + -30 = 0 -30 + 18a + -2a2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-15 + 9a + -1a2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-15 + 9a + -1a2)' equal to zero and attempt to solve: Simplifying -15 + 9a + -1a2 = 0 Solving -15 + 9a + -1a2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 15 + -9a + a2 = 0 Move the constant term to the right: Add '-15' to each side of the equation. 15 + -9a + -15 + a2 = 0 + -15 Reorder the terms: 15 + -15 + -9a + a2 = 0 + -15 Combine like terms: 15 + -15 = 0 0 + -9a + a2 = 0 + -15 -9a + a2 = 0 + -15 Combine like terms: 0 + -15 = -15 -9a + a2 = -15 The a term is -9a. Take half its coefficient (-4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. -9a + 20.25 + a2 = -15 + 20.25 Reorder the terms: 20.25 + -9a + a2 = -15 + 20.25 Combine like terms: -15 + 20.25 = 5.25 20.25 + -9a + a2 = 5.25 Factor a perfect square on the left side: (a + -4.5)(a + -4.5) = 5.25 Calculate the square root of the right side: 2.291287847 Break this problem into two subproblems by setting (a + -4.5) equal to 2.291287847 and -2.291287847.Subproblem 1
a + -4.5 = 2.291287847 Simplifying a + -4.5 = 2.291287847 Reorder the terms: -4.5 + a = 2.291287847 Solving -4.5 + a = 2.291287847 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + a = 2.291287847 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + a = 2.291287847 + 4.5 a = 2.291287847 + 4.5 Combine like terms: 2.291287847 + 4.5 = 6.791287847 a = 6.791287847 Simplifying a = 6.791287847Subproblem 2
a + -4.5 = -2.291287847 Simplifying a + -4.5 = -2.291287847 Reorder the terms: -4.5 + a = -2.291287847 Solving -4.5 + a = -2.291287847 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + a = -2.291287847 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + a = -2.291287847 + 4.5 a = -2.291287847 + 4.5 Combine like terms: -2.291287847 + 4.5 = 2.208712153 a = 2.208712153 Simplifying a = 2.208712153Solution
The solution to the problem is based on the solutions from the subproblems. a = {6.791287847, 2.208712153}Solution
a = {6.791287847, 2.208712153}
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