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Simplifying 5a3 + -25a2 + 15a = 0 Reorder the terms: 15a + -25a2 + 5a3 = 0 Solving 15a + -25a2 + 5a3 = 0 Solving for variable 'a'. Factor out the Greatest Common Factor (GCF), '5a'. 5a(3 + -5a + a2) = 0 Ignore the factor 5.Subproblem 1
Set the factor 'a' equal to zero and attempt to solve: Simplifying a = 0 Solving a = 0 Move all terms containing a to the left, all other terms to the right. Simplifying a = 0Subproblem 2
Set the factor '(3 + -5a + a2)' equal to zero and attempt to solve: Simplifying 3 + -5a + a2 = 0 Solving 3 + -5a + a2 = 0 Begin completing the square. Move the constant term to the right: Add '-3' to each side of the equation. 3 + -5a + -3 + a2 = 0 + -3 Reorder the terms: 3 + -3 + -5a + a2 = 0 + -3 Combine like terms: 3 + -3 = 0 0 + -5a + a2 = 0 + -3 -5a + a2 = 0 + -3 Combine like terms: 0 + -3 = -3 -5a + a2 = -3 The a term is -5a. Take half its coefficient (-2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. -5a + 6.25 + a2 = -3 + 6.25 Reorder the terms: 6.25 + -5a + a2 = -3 + 6.25 Combine like terms: -3 + 6.25 = 3.25 6.25 + -5a + a2 = 3.25 Factor a perfect square on the left side: (a + -2.5)(a + -2.5) = 3.25 Calculate the square root of the right side: 1.802775638 Break this problem into two subproblems by setting (a + -2.5) equal to 1.802775638 and -1.802775638.Subproblem 1
a + -2.5 = 1.802775638 Simplifying a + -2.5 = 1.802775638 Reorder the terms: -2.5 + a = 1.802775638 Solving -2.5 + a = 1.802775638 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + a = 1.802775638 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + a = 1.802775638 + 2.5 a = 1.802775638 + 2.5 Combine like terms: 1.802775638 + 2.5 = 4.302775638 a = 4.302775638 Simplifying a = 4.302775638Subproblem 2
a + -2.5 = -1.802775638 Simplifying a + -2.5 = -1.802775638 Reorder the terms: -2.5 + a = -1.802775638 Solving -2.5 + a = -1.802775638 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + a = -1.802775638 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + a = -1.802775638 + 2.5 a = -1.802775638 + 2.5 Combine like terms: -1.802775638 + 2.5 = 0.697224362 a = 0.697224362 Simplifying a = 0.697224362Solution
The solution to the problem is based on the solutions from the subproblems. a = {4.302775638, 0.697224362}Solution
a = {0, 4.302775638, 0.697224362}
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