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Simplifying 10a2 + 10a + -8 = 0 Reorder the terms: -8 + 10a + 10a2 = 0 Solving -8 + 10a + 10a2 = 0 Solving for variable 'a'. Factor out the Greatest Common Factor (GCF), '2'. 2(-4 + 5a + 5a2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-4 + 5a + 5a2)' equal to zero and attempt to solve: Simplifying -4 + 5a + 5a2 = 0 Solving -4 + 5a + 5a2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -0.8 + a + a2 = 0 Move the constant term to the right: Add '0.8' to each side of the equation. -0.8 + a + 0.8 + a2 = 0 + 0.8 Reorder the terms: -0.8 + 0.8 + a + a2 = 0 + 0.8 Combine like terms: -0.8 + 0.8 = 0.0 0.0 + a + a2 = 0 + 0.8 a + a2 = 0 + 0.8 Combine like terms: 0 + 0.8 = 0.8 a + a2 = 0.8 The a term is a. 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. + 0.25 + a2 = 0.8 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + a2 = 0.8 + 0.25 Combine like terms: 0.8 + 0.25 = 1.05 1.25 + a2 = 1.05 Factor a perfect square on the left side: (a + 0.5)(a + 0.5) = 1.05 Calculate the square root of the right side: 1.024695077 Break this problem into two subproblems by setting (a + 0.5) equal to 1.024695077 and -1.024695077.Subproblem 1
a + 0.5 = 1.024695077 Simplifying a + 0.5 = 1.024695077 Reorder the terms: 0.5 + a = 1.024695077 Solving 0.5 + a = 1.024695077 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 = 1.024695077 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = 1.024695077 + -0.5 a = 1.024695077 + -0.5 Combine like terms: 1.024695077 + -0.5 = 0.524695077 a = 0.524695077 Simplifying a = 0.524695077Subproblem 2
a + 0.5 = -1.024695077 Simplifying a + 0.5 = -1.024695077 Reorder the terms: 0.5 + a = -1.024695077 Solving 0.5 + a = -1.024695077 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 = -1.024695077 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = -1.024695077 + -0.5 a = -1.024695077 + -0.5 Combine like terms: -1.024695077 + -0.5 = -1.524695077 a = -1.524695077 Simplifying a = -1.524695077Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.524695077, -1.524695077}Solution
a = {0.524695077, -1.524695077}
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