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Simplifying 2a2(5a + -6) + -5a(a2 + -3a + 4) = 0 Reorder the terms: 2a2(-6 + 5a) + -5a(a2 + -3a + 4) = 0 (-6 * 2a2 + 5a * 2a2) + -5a(a2 + -3a + 4) = 0 (-12a2 + 10a3) + -5a(a2 + -3a + 4) = 0 Reorder the terms: -12a2 + 10a3 + -5a(4 + -3a + a2) = 0 -12a2 + 10a3 + (4 * -5a + -3a * -5a + a2 * -5a) = 0 -12a2 + 10a3 + (-20a + 15a2 + -5a3) = 0 Reorder the terms: -20a + -12a2 + 15a2 + 10a3 + -5a3 = 0 Combine like terms: -12a2 + 15a2 = 3a2 -20a + 3a2 + 10a3 + -5a3 = 0 Combine like terms: 10a3 + -5a3 = 5a3 -20a + 3a2 + 5a3 = 0 Solving -20a + 3a2 + 5a3 = 0 Solving for variable 'a'. Factor out the Greatest Common Factor (GCF), 'a'. a(-20 + 3a + 5a2) = 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 '(-20 + 3a + 5a2)' equal to zero and attempt to solve: Simplifying -20 + 3a + 5a2 = 0 Solving -20 + 3a + 5a2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -4 + 0.6a + a2 = 0 Move the constant term to the right: Add '4' to each side of the equation. -4 + 0.6a + 4 + a2 = 0 + 4 Reorder the terms: -4 + 4 + 0.6a + a2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + 0.6a + a2 = 0 + 4 0.6a + a2 = 0 + 4 Combine like terms: 0 + 4 = 4 0.6a + a2 = 4 The a term is 0.6a. Take half its coefficient (0.3). Square it (0.09) and add it to both sides. Add '0.09' to each side of the equation. 0.6a + 0.09 + a2 = 4 + 0.09 Reorder the terms: 0.09 + 0.6a + a2 = 4 + 0.09 Combine like terms: 4 + 0.09 = 4.09 0.09 + 0.6a + a2 = 4.09 Factor a perfect square on the left side: (a + 0.3)(a + 0.3) = 4.09 Calculate the square root of the right side: 2.022374842 Break this problem into two subproblems by setting (a + 0.3) equal to 2.022374842 and -2.022374842.Subproblem 1
a + 0.3 = 2.022374842 Simplifying a + 0.3 = 2.022374842 Reorder the terms: 0.3 + a = 2.022374842 Solving 0.3 + a = 2.022374842 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.3' to each side of the equation. 0.3 + -0.3 + a = 2.022374842 + -0.3 Combine like terms: 0.3 + -0.3 = 0.0 0.0 + a = 2.022374842 + -0.3 a = 2.022374842 + -0.3 Combine like terms: 2.022374842 + -0.3 = 1.722374842 a = 1.722374842 Simplifying a = 1.722374842Subproblem 2
a + 0.3 = -2.022374842 Simplifying a + 0.3 = -2.022374842 Reorder the terms: 0.3 + a = -2.022374842 Solving 0.3 + a = -2.022374842 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.3' to each side of the equation. 0.3 + -0.3 + a = -2.022374842 + -0.3 Combine like terms: 0.3 + -0.3 = 0.0 0.0 + a = -2.022374842 + -0.3 a = -2.022374842 + -0.3 Combine like terms: -2.022374842 + -0.3 = -2.322374842 a = -2.322374842 Simplifying a = -2.322374842Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.722374842, -2.322374842}Solution
a = {0, 1.722374842, -2.322374842}
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