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Simplifying 42a2 + -24a + -72 = 0 Reorder the terms: -72 + -24a + 42a2 = 0 Solving -72 + -24a + 42a2 = 0 Solving for variable 'a'. Factor out the Greatest Common Factor (GCF), '6'. 6(-12 + -4a + 7a2) = 0 Ignore the factor 6.Subproblem 1
Set the factor '(-12 + -4a + 7a2)' equal to zero and attempt to solve: Simplifying -12 + -4a + 7a2 = 0 Solving -12 + -4a + 7a2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -1.714285714 + -0.5714285714a + a2 = 0 Move the constant term to the right: Add '1.714285714' to each side of the equation. -1.714285714 + -0.5714285714a + 1.714285714 + a2 = 0 + 1.714285714 Reorder the terms: -1.714285714 + 1.714285714 + -0.5714285714a + a2 = 0 + 1.714285714 Combine like terms: -1.714285714 + 1.714285714 = 0.000000000 0.000000000 + -0.5714285714a + a2 = 0 + 1.714285714 -0.5714285714a + a2 = 0 + 1.714285714 Combine like terms: 0 + 1.714285714 = 1.714285714 -0.5714285714a + a2 = 1.714285714 The a term is -0.5714285714a. Take half its coefficient (-0.2857142857). Square it (0.08163265305) and add it to both sides. Add '0.08163265305' to each side of the equation. -0.5714285714a + 0.08163265305 + a2 = 1.714285714 + 0.08163265305 Reorder the terms: 0.08163265305 + -0.5714285714a + a2 = 1.714285714 + 0.08163265305 Combine like terms: 1.714285714 + 0.08163265305 = 1.79591836705 0.08163265305 + -0.5714285714a + a2 = 1.79591836705 Factor a perfect square on the left side: (a + -0.2857142857)(a + -0.2857142857) = 1.79591836705 Calculate the square root of the right side: 1.340118788 Break this problem into two subproblems by setting (a + -0.2857142857) equal to 1.340118788 and -1.340118788.Subproblem 1
a + -0.2857142857 = 1.340118788 Simplifying a + -0.2857142857 = 1.340118788 Reorder the terms: -0.2857142857 + a = 1.340118788 Solving -0.2857142857 + a = 1.340118788 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.2857142857' to each side of the equation. -0.2857142857 + 0.2857142857 + a = 1.340118788 + 0.2857142857 Combine like terms: -0.2857142857 + 0.2857142857 = 0.0000000000 0.0000000000 + a = 1.340118788 + 0.2857142857 a = 1.340118788 + 0.2857142857 Combine like terms: 1.340118788 + 0.2857142857 = 1.6258330737 a = 1.6258330737 Simplifying a = 1.6258330737Subproblem 2
a + -0.2857142857 = -1.340118788 Simplifying a + -0.2857142857 = -1.340118788 Reorder the terms: -0.2857142857 + a = -1.340118788 Solving -0.2857142857 + a = -1.340118788 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.2857142857' to each side of the equation. -0.2857142857 + 0.2857142857 + a = -1.340118788 + 0.2857142857 Combine like terms: -0.2857142857 + 0.2857142857 = 0.0000000000 0.0000000000 + a = -1.340118788 + 0.2857142857 a = -1.340118788 + 0.2857142857 Combine like terms: -1.340118788 + 0.2857142857 = -1.0544045023 a = -1.0544045023 Simplifying a = -1.0544045023Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.6258330737, -1.0544045023}Solution
a = {1.6258330737, -1.0544045023}
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