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Simplifying 6a(3a + -2) + 5a = 57 Reorder the terms: 6a(-2 + 3a) + 5a = 57 (-2 * 6a + 3a * 6a) + 5a = 57 (-12a + 18a2) + 5a = 57 Reorder the terms: -12a + 5a + 18a2 = 57 Combine like terms: -12a + 5a = -7a -7a + 18a2 = 57 Solving -7a + 18a2 = 57 Solving for variable 'a'. Reorder the terms: -57 + -7a + 18a2 = 57 + -57 Combine like terms: 57 + -57 = 0 -57 + -7a + 18a2 = 0 Begin completing the square. Divide all terms by 18 the coefficient of the squared term: Divide each side by '18'. -3.166666667 + -0.3888888889a + a2 = 0 Move the constant term to the right: Add '3.166666667' to each side of the equation. -3.166666667 + -0.3888888889a + 3.166666667 + a2 = 0 + 3.166666667 Reorder the terms: -3.166666667 + 3.166666667 + -0.3888888889a + a2 = 0 + 3.166666667 Combine like terms: -3.166666667 + 3.166666667 = 0.000000000 0.000000000 + -0.3888888889a + a2 = 0 + 3.166666667 -0.3888888889a + a2 = 0 + 3.166666667 Combine like terms: 0 + 3.166666667 = 3.166666667 -0.3888888889a + a2 = 3.166666667 The a term is -0.3888888889a. Take half its coefficient (-0.1944444445). Square it (0.03780864200) and add it to both sides. Add '0.03780864200' to each side of the equation. -0.3888888889a + 0.03780864200 + a2 = 3.166666667 + 0.03780864200 Reorder the terms: 0.03780864200 + -0.3888888889a + a2 = 3.166666667 + 0.03780864200 Combine like terms: 3.166666667 + 0.03780864200 = 3.204475309 0.03780864200 + -0.3888888889a + a2 = 3.204475309 Factor a perfect square on the left side: (a + -0.1944444445)(a + -0.1944444445) = 3.204475309 Calculate the square root of the right side: 1.790104832 Break this problem into two subproblems by setting (a + -0.1944444445) equal to 1.790104832 and -1.790104832.Subproblem 1
a + -0.1944444445 = 1.790104832 Simplifying a + -0.1944444445 = 1.790104832 Reorder the terms: -0.1944444445 + a = 1.790104832 Solving -0.1944444445 + a = 1.790104832 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.1944444445' to each side of the equation. -0.1944444445 + 0.1944444445 + a = 1.790104832 + 0.1944444445 Combine like terms: -0.1944444445 + 0.1944444445 = 0.0000000000 0.0000000000 + a = 1.790104832 + 0.1944444445 a = 1.790104832 + 0.1944444445 Combine like terms: 1.790104832 + 0.1944444445 = 1.9845492765 a = 1.9845492765 Simplifying a = 1.9845492765Subproblem 2
a + -0.1944444445 = -1.790104832 Simplifying a + -0.1944444445 = -1.790104832 Reorder the terms: -0.1944444445 + a = -1.790104832 Solving -0.1944444445 + a = -1.790104832 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.1944444445' to each side of the equation. -0.1944444445 + 0.1944444445 + a = -1.790104832 + 0.1944444445 Combine like terms: -0.1944444445 + 0.1944444445 = 0.0000000000 0.0000000000 + a = -1.790104832 + 0.1944444445 a = -1.790104832 + 0.1944444445 Combine like terms: -1.790104832 + 0.1944444445 = -1.5956603875 a = -1.5956603875 Simplifying a = -1.5956603875Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.9845492765, -1.5956603875}
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