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Simplifying 14a4 + 7a3 + -10a2 = 0 Reorder the terms: -10a2 + 7a3 + 14a4 = 0 Solving -10a2 + 7a3 + 14a4 = 0 Solving for variable 'a'. Factor out the Greatest Common Factor (GCF), 'a2'. a2(-10 + 7a + 14a2) = 0Subproblem 1
Set the factor 'a2' equal to zero and attempt to solve: Simplifying a2 = 0 Solving a2 = 0 Move all terms containing a to the left, all other terms to the right. Simplifying a2 = 0 Take the square root of each side: a = {0}Subproblem 2
Set the factor '(-10 + 7a + 14a2)' equal to zero and attempt to solve: Simplifying -10 + 7a + 14a2 = 0 Solving -10 + 7a + 14a2 = 0 Begin completing the square. Divide all terms by 14 the coefficient of the squared term: Divide each side by '14'. -0.7142857143 + 0.5a + a2 = 0 Move the constant term to the right: Add '0.7142857143' to each side of the equation. -0.7142857143 + 0.5a + 0.7142857143 + a2 = 0 + 0.7142857143 Reorder the terms: -0.7142857143 + 0.7142857143 + 0.5a + a2 = 0 + 0.7142857143 Combine like terms: -0.7142857143 + 0.7142857143 = 0.0000000000 0.0000000000 + 0.5a + a2 = 0 + 0.7142857143 0.5a + a2 = 0 + 0.7142857143 Combine like terms: 0 + 0.7142857143 = 0.7142857143 0.5a + a2 = 0.7142857143 The a term is 0.5a. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5a + 0.0625 + a2 = 0.7142857143 + 0.0625 Reorder the terms: 0.0625 + 0.5a + a2 = 0.7142857143 + 0.0625 Combine like terms: 0.7142857143 + 0.0625 = 0.7767857143 0.0625 + 0.5a + a2 = 0.7767857143 Factor a perfect square on the left side: (a + 0.25)(a + 0.25) = 0.7767857143 Calculate the square root of the right side: 0.881354477 Break this problem into two subproblems by setting (a + 0.25) equal to 0.881354477 and -0.881354477.Subproblem 1
a + 0.25 = 0.881354477 Simplifying a + 0.25 = 0.881354477 Reorder the terms: 0.25 + a = 0.881354477 Solving 0.25 + a = 0.881354477 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + a = 0.881354477 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + a = 0.881354477 + -0.25 a = 0.881354477 + -0.25 Combine like terms: 0.881354477 + -0.25 = 0.631354477 a = 0.631354477 Simplifying a = 0.631354477Subproblem 2
a + 0.25 = -0.881354477 Simplifying a + 0.25 = -0.881354477 Reorder the terms: 0.25 + a = -0.881354477 Solving 0.25 + a = -0.881354477 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + a = -0.881354477 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + a = -0.881354477 + -0.25 a = -0.881354477 + -0.25 Combine like terms: -0.881354477 + -0.25 = -1.131354477 a = -1.131354477 Simplifying a = -1.131354477Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.631354477, -1.131354477}Solution
a = {0, 0.631354477, -1.131354477}
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