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Simplifying 34a2 + -832a + 2314 = 0 Reorder the terms: 2314 + -832a + 34a2 = 0 Solving 2314 + -832a + 34a2 = 0 Solving for variable 'a'. Factor out the Greatest Common Factor (GCF), '2'. 2(1157 + -416a + 17a2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(1157 + -416a + 17a2)' equal to zero and attempt to solve: Simplifying 1157 + -416a + 17a2 = 0 Solving 1157 + -416a + 17a2 = 0 Begin completing the square. Divide all terms by 17 the coefficient of the squared term: Divide each side by '17'. 68.05882353 + -24.47058824a + a2 = 0 Move the constant term to the right: Add '-68.05882353' to each side of the equation. 68.05882353 + -24.47058824a + -68.05882353 + a2 = 0 + -68.05882353 Reorder the terms: 68.05882353 + -68.05882353 + -24.47058824a + a2 = 0 + -68.05882353 Combine like terms: 68.05882353 + -68.05882353 = 0.00000000 0.00000000 + -24.47058824a + a2 = 0 + -68.05882353 -24.47058824a + a2 = 0 + -68.05882353 Combine like terms: 0 + -68.05882353 = -68.05882353 -24.47058824a + a2 = -68.05882353 The a term is -24.47058824a. Take half its coefficient (-12.23529412). Square it (149.7024222) and add it to both sides. Add '149.7024222' to each side of the equation. -24.47058824a + 149.7024222 + a2 = -68.05882353 + 149.7024222 Reorder the terms: 149.7024222 + -24.47058824a + a2 = -68.05882353 + 149.7024222 Combine like terms: -68.05882353 + 149.7024222 = 81.64359867 149.7024222 + -24.47058824a + a2 = 81.64359867 Factor a perfect square on the left side: (a + -12.23529412)(a + -12.23529412) = 81.64359867 Calculate the square root of the right side: 9.035684737 Break this problem into two subproblems by setting (a + -12.23529412) equal to 9.035684737 and -9.035684737.Subproblem 1
a + -12.23529412 = 9.035684737 Simplifying a + -12.23529412 = 9.035684737 Reorder the terms: -12.23529412 + a = 9.035684737 Solving -12.23529412 + a = 9.035684737 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '12.23529412' to each side of the equation. -12.23529412 + 12.23529412 + a = 9.035684737 + 12.23529412 Combine like terms: -12.23529412 + 12.23529412 = 0.00000000 0.00000000 + a = 9.035684737 + 12.23529412 a = 9.035684737 + 12.23529412 Combine like terms: 9.035684737 + 12.23529412 = 21.270978857 a = 21.270978857 Simplifying a = 21.270978857Subproblem 2
a + -12.23529412 = -9.035684737 Simplifying a + -12.23529412 = -9.035684737 Reorder the terms: -12.23529412 + a = -9.035684737 Solving -12.23529412 + a = -9.035684737 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '12.23529412' to each side of the equation. -12.23529412 + 12.23529412 + a = -9.035684737 + 12.23529412 Combine like terms: -12.23529412 + 12.23529412 = 0.00000000 0.00000000 + a = -9.035684737 + 12.23529412 a = -9.035684737 + 12.23529412 Combine like terms: -9.035684737 + 12.23529412 = 3.199609383 a = 3.199609383 Simplifying a = 3.199609383Solution
The solution to the problem is based on the solutions from the subproblems. a = {21.270978857, 3.199609383}Solution
a = {21.270978857, 3.199609383}
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