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Simplifying 10a + 5a(a + -3) = 15 Reorder the terms: 10a + 5a(-3 + a) = 15 10a + (-3 * 5a + a * 5a) = 15 10a + (-15a + 5a2) = 15 Combine like terms: 10a + -15a = -5a -5a + 5a2 = 15 Solving -5a + 5a2 = 15 Solving for variable 'a'. Reorder the terms: -15 + -5a + 5a2 = 15 + -15 Combine like terms: 15 + -15 = 0 -15 + -5a + 5a2 = 0 Factor out the Greatest Common Factor (GCF), '5'. 5(-3 + -1a + a2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(-3 + -1a + a2)' equal to zero and attempt to solve: Simplifying -3 + -1a + a2 = 0 Solving -3 + -1a + a2 = 0 Begin completing the square. Move the constant term to the right: Add '3' to each side of the equation. -3 + -1a + 3 + a2 = 0 + 3 Reorder the terms: -3 + 3 + -1a + a2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + -1a + a2 = 0 + 3 -1a + a2 = 0 + 3 Combine like terms: 0 + 3 = 3 -1a + a2 = 3 The a term is -1a. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1a + 0.25 + a2 = 3 + 0.25 Reorder the terms: 0.25 + -1a + a2 = 3 + 0.25 Combine like terms: 3 + 0.25 = 3.25 0.25 + -1a + a2 = 3.25 Factor a perfect square on the left side: (a + -0.5)(a + -0.5) = 3.25 Calculate the square root of the right side: 1.802775638 Break this problem into two subproblems by setting (a + -0.5) equal to 1.802775638 and -1.802775638.Subproblem 1
a + -0.5 = 1.802775638 Simplifying a + -0.5 = 1.802775638 Reorder the terms: -0.5 + a = 1.802775638 Solving -0.5 + a = 1.802775638 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + a = 1.802775638 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + a = 1.802775638 + 0.5 a = 1.802775638 + 0.5 Combine like terms: 1.802775638 + 0.5 = 2.302775638 a = 2.302775638 Simplifying a = 2.302775638Subproblem 2
a + -0.5 = -1.802775638 Simplifying a + -0.5 = -1.802775638 Reorder the terms: -0.5 + a = -1.802775638 Solving -0.5 + a = -1.802775638 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + a = -1.802775638 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + a = -1.802775638 + 0.5 a = -1.802775638 + 0.5 Combine like terms: -1.802775638 + 0.5 = -1.302775638 a = -1.302775638 Simplifying a = -1.302775638Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.302775638, -1.302775638}Solution
a = {2.302775638, -1.302775638}
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