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Simplifying 5a(7a + -3) = 35 Reorder the terms: 5a(-3 + 7a) = 35 (-3 * 5a + 7a * 5a) = 35 (-15a + 35a2) = 35 Solving -15a + 35a2 = 35 Solving for variable 'a'. Reorder the terms: -35 + -15a + 35a2 = 35 + -35 Combine like terms: 35 + -35 = 0 -35 + -15a + 35a2 = 0 Factor out the Greatest Common Factor (GCF), '5'. 5(-7 + -3a + 7a2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(-7 + -3a + 7a2)' equal to zero and attempt to solve: Simplifying -7 + -3a + 7a2 = 0 Solving -7 + -3a + 7a2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -1 + -0.4285714286a + a2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -0.4285714286a + 1 + a2 = 0 + 1 Reorder the terms: -1 + 1 + -0.4285714286a + a2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -0.4285714286a + a2 = 0 + 1 -0.4285714286a + a2 = 0 + 1 Combine like terms: 0 + 1 = 1 -0.4285714286a + a2 = 1 The a term is -0.4285714286a. Take half its coefficient (-0.2142857143). Square it (0.04591836735) and add it to both sides. Add '0.04591836735' to each side of the equation. -0.4285714286a + 0.04591836735 + a2 = 1 + 0.04591836735 Reorder the terms: 0.04591836735 + -0.4285714286a + a2 = 1 + 0.04591836735 Combine like terms: 1 + 0.04591836735 = 1.04591836735 0.04591836735 + -0.4285714286a + a2 = 1.04591836735 Factor a perfect square on the left side: (a + -0.2142857143)(a + -0.2142857143) = 1.04591836735 Calculate the square root of the right side: 1.022701505 Break this problem into two subproblems by setting (a + -0.2142857143) equal to 1.022701505 and -1.022701505.Subproblem 1
a + -0.2142857143 = 1.022701505 Simplifying a + -0.2142857143 = 1.022701505 Reorder the terms: -0.2142857143 + a = 1.022701505 Solving -0.2142857143 + a = 1.022701505 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.2142857143' to each side of the equation. -0.2142857143 + 0.2142857143 + a = 1.022701505 + 0.2142857143 Combine like terms: -0.2142857143 + 0.2142857143 = 0.0000000000 0.0000000000 + a = 1.022701505 + 0.2142857143 a = 1.022701505 + 0.2142857143 Combine like terms: 1.022701505 + 0.2142857143 = 1.2369872193 a = 1.2369872193 Simplifying a = 1.2369872193Subproblem 2
a + -0.2142857143 = -1.022701505 Simplifying a + -0.2142857143 = -1.022701505 Reorder the terms: -0.2142857143 + a = -1.022701505 Solving -0.2142857143 + a = -1.022701505 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.2142857143' to each side of the equation. -0.2142857143 + 0.2142857143 + a = -1.022701505 + 0.2142857143 Combine like terms: -0.2142857143 + 0.2142857143 = 0.0000000000 0.0000000000 + a = -1.022701505 + 0.2142857143 a = -1.022701505 + 0.2142857143 Combine like terms: -1.022701505 + 0.2142857143 = -0.8084157907 a = -0.8084157907 Simplifying a = -0.8084157907Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.2369872193, -0.8084157907}Solution
a = {1.2369872193, -0.8084157907}
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