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Simplifying -7a2 + 65a + 18 = 0 Reorder the terms: 18 + 65a + -7a2 = 0 Solving 18 + 65a + -7a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by -7 the coefficient of the squared term: Divide each side by '-7'. -2.571428571 + -9.285714286a + a2 = 0 Move the constant term to the right: Add '2.571428571' to each side of the equation. -2.571428571 + -9.285714286a + 2.571428571 + a2 = 0 + 2.571428571 Reorder the terms: -2.571428571 + 2.571428571 + -9.285714286a + a2 = 0 + 2.571428571 Combine like terms: -2.571428571 + 2.571428571 = 0.000000000 0.000000000 + -9.285714286a + a2 = 0 + 2.571428571 -9.285714286a + a2 = 0 + 2.571428571 Combine like terms: 0 + 2.571428571 = 2.571428571 -9.285714286a + a2 = 2.571428571 The a term is -9.285714286a. Take half its coefficient (-4.642857143). Square it (21.55612245) and add it to both sides. Add '21.55612245' to each side of the equation. -9.285714286a + 21.55612245 + a2 = 2.571428571 + 21.55612245 Reorder the terms: 21.55612245 + -9.285714286a + a2 = 2.571428571 + 21.55612245 Combine like terms: 2.571428571 + 21.55612245 = 24.127551021 21.55612245 + -9.285714286a + a2 = 24.127551021 Factor a perfect square on the left side: (a + -4.642857143)(a + -4.642857143) = 24.127551021 Calculate the square root of the right side: 4.911980356 Break this problem into two subproblems by setting (a + -4.642857143) equal to 4.911980356 and -4.911980356.Subproblem 1
a + -4.642857143 = 4.911980356 Simplifying a + -4.642857143 = 4.911980356 Reorder the terms: -4.642857143 + a = 4.911980356 Solving -4.642857143 + a = 4.911980356 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '4.642857143' to each side of the equation. -4.642857143 + 4.642857143 + a = 4.911980356 + 4.642857143 Combine like terms: -4.642857143 + 4.642857143 = 0.000000000 0.000000000 + a = 4.911980356 + 4.642857143 a = 4.911980356 + 4.642857143 Combine like terms: 4.911980356 + 4.642857143 = 9.554837499 a = 9.554837499 Simplifying a = 9.554837499Subproblem 2
a + -4.642857143 = -4.911980356 Simplifying a + -4.642857143 = -4.911980356 Reorder the terms: -4.642857143 + a = -4.911980356 Solving -4.642857143 + a = -4.911980356 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '4.642857143' to each side of the equation. -4.642857143 + 4.642857143 + a = -4.911980356 + 4.642857143 Combine like terms: -4.642857143 + 4.642857143 = 0.000000000 0.000000000 + a = -4.911980356 + 4.642857143 a = -4.911980356 + 4.642857143 Combine like terms: -4.911980356 + 4.642857143 = -0.269123213 a = -0.269123213 Simplifying a = -0.269123213Solution
The solution to the problem is based on the solutions from the subproblems. a = {9.554837499, -0.269123213}
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