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Simplifying 30a2 + 12a + -21 = 0 Reorder the terms: -21 + 12a + 30a2 = 0 Solving -21 + 12a + 30a2 = 0 Solving for variable 'a'. Factor out the Greatest Common Factor (GCF), '3'. 3(-7 + 4a + 10a2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-7 + 4a + 10a2)' equal to zero and attempt to solve: Simplifying -7 + 4a + 10a2 = 0 Solving -7 + 4a + 10a2 = 0 Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. -0.7 + 0.4a + a2 = 0.0 Move the constant term to the right: Add '0.7' to each side of the equation. -0.7 + 0.4a + 0.7 + a2 = 0.0 + 0.7 Reorder the terms: -0.7 + 0.7 + 0.4a + a2 = 0.0 + 0.7 Combine like terms: -0.7 + 0.7 = 0.0 0.0 + 0.4a + a2 = 0.0 + 0.7 0.4a + a2 = 0.0 + 0.7 Combine like terms: 0.0 + 0.7 = 0.7 0.4a + a2 = 0.7 The a term is 0.4a. Take half its coefficient (0.2). Square it (0.04) and add it to both sides. Add '0.04' to each side of the equation. 0.4a + 0.04 + a2 = 0.7 + 0.04 Reorder the terms: 0.04 + 0.4a + a2 = 0.7 + 0.04 Combine like terms: 0.7 + 0.04 = 0.74 0.04 + 0.4a + a2 = 0.74 Factor a perfect square on the left side: (a + 0.2)(a + 0.2) = 0.74 Calculate the square root of the right side: 0.860232527 Break this problem into two subproblems by setting (a + 0.2) equal to 0.860232527 and -0.860232527.Subproblem 1
a + 0.2 = 0.860232527 Simplifying a + 0.2 = 0.860232527 Reorder the terms: 0.2 + a = 0.860232527 Solving 0.2 + a = 0.860232527 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.2' to each side of the equation. 0.2 + -0.2 + a = 0.860232527 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + a = 0.860232527 + -0.2 a = 0.860232527 + -0.2 Combine like terms: 0.860232527 + -0.2 = 0.660232527 a = 0.660232527 Simplifying a = 0.660232527Subproblem 2
a + 0.2 = -0.860232527 Simplifying a + 0.2 = -0.860232527 Reorder the terms: 0.2 + a = -0.860232527 Solving 0.2 + a = -0.860232527 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.2' to each side of the equation. 0.2 + -0.2 + a = -0.860232527 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + a = -0.860232527 + -0.2 a = -0.860232527 + -0.2 Combine like terms: -0.860232527 + -0.2 = -1.060232527 a = -1.060232527 Simplifying a = -1.060232527Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.660232527, -1.060232527}Solution
a = {0.660232527, -1.060232527}
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