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Simplifying 5a(3a2 + -2a + -6) + -4a(a + -3) = 0 Reorder the terms: 5a(-6 + -2a + 3a2) + -4a(a + -3) = 0 (-6 * 5a + -2a * 5a + 3a2 * 5a) + -4a(a + -3) = 0 (-30a + -10a2 + 15a3) + -4a(a + -3) = 0 Reorder the terms: -30a + -10a2 + 15a3 + -4a(-3 + a) = 0 -30a + -10a2 + 15a3 + (-3 * -4a + a * -4a) = 0 -30a + -10a2 + 15a3 + (12a + -4a2) = 0 Reorder the terms: -30a + 12a + -10a2 + -4a2 + 15a3 = 0 Combine like terms: -30a + 12a = -18a -18a + -10a2 + -4a2 + 15a3 = 0 Combine like terms: -10a2 + -4a2 = -14a2 -18a + -14a2 + 15a3 = 0 Solving -18a + -14a2 + 15a3 = 0 Solving for variable 'a'. Factor out the Greatest Common Factor (GCF), 'a'. a(-18 + -14a + 15a2) = 0Subproblem 1
Set the factor 'a' equal to zero and attempt to solve: Simplifying a = 0 Solving a = 0 Move all terms containing a to the left, all other terms to the right. Simplifying a = 0Subproblem 2
Set the factor '(-18 + -14a + 15a2)' equal to zero and attempt to solve: Simplifying -18 + -14a + 15a2 = 0 Solving -18 + -14a + 15a2 = 0 Begin completing the square. Divide all terms by 15 the coefficient of the squared term: Divide each side by '15'. -1.2 + -0.9333333333a + a2 = 0 Move the constant term to the right: Add '1.2' to each side of the equation. -1.2 + -0.9333333333a + 1.2 + a2 = 0 + 1.2 Reorder the terms: -1.2 + 1.2 + -0.9333333333a + a2 = 0 + 1.2 Combine like terms: -1.2 + 1.2 = 0.0 0.0 + -0.9333333333a + a2 = 0 + 1.2 -0.9333333333a + a2 = 0 + 1.2 Combine like terms: 0 + 1.2 = 1.2 -0.9333333333a + a2 = 1.2 The a term is -0.9333333333a. Take half its coefficient (-0.4666666667). Square it (0.2177777778) and add it to both sides. Add '0.2177777778' to each side of the equation. -0.9333333333a + 0.2177777778 + a2 = 1.2 + 0.2177777778 Reorder the terms: 0.2177777778 + -0.9333333333a + a2 = 1.2 + 0.2177777778 Combine like terms: 1.2 + 0.2177777778 = 1.4177777778 0.2177777778 + -0.9333333333a + a2 = 1.4177777778 Factor a perfect square on the left side: (a + -0.4666666667)(a + -0.4666666667) = 1.4177777778 Calculate the square root of the right side: 1.19070474 Break this problem into two subproblems by setting (a + -0.4666666667) equal to 1.19070474 and -1.19070474.Subproblem 1
a + -0.4666666667 = 1.19070474 Simplifying a + -0.4666666667 = 1.19070474 Reorder the terms: -0.4666666667 + a = 1.19070474 Solving -0.4666666667 + a = 1.19070474 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.4666666667' to each side of the equation. -0.4666666667 + 0.4666666667 + a = 1.19070474 + 0.4666666667 Combine like terms: -0.4666666667 + 0.4666666667 = 0.0000000000 0.0000000000 + a = 1.19070474 + 0.4666666667 a = 1.19070474 + 0.4666666667 Combine like terms: 1.19070474 + 0.4666666667 = 1.6573714067 a = 1.6573714067 Simplifying a = 1.6573714067Subproblem 2
a + -0.4666666667 = -1.19070474 Simplifying a + -0.4666666667 = -1.19070474 Reorder the terms: -0.4666666667 + a = -1.19070474 Solving -0.4666666667 + a = -1.19070474 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.4666666667' to each side of the equation. -0.4666666667 + 0.4666666667 + a = -1.19070474 + 0.4666666667 Combine like terms: -0.4666666667 + 0.4666666667 = 0.0000000000 0.0000000000 + a = -1.19070474 + 0.4666666667 a = -1.19070474 + 0.4666666667 Combine like terms: -1.19070474 + 0.4666666667 = -0.7240380733 a = -0.7240380733 Simplifying a = -0.7240380733Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.6573714067, -0.7240380733}Solution
a = {0, 1.6573714067, -0.7240380733}
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