18+65a-7a^2=0

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Solution for 18+65a-7a^2=0 equation:


Simplifying
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 = 9.554837499

Subproblem 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 = -0.269123213

Solution

The solution to the problem is based on the solutions from the subproblems. a = {9.554837499, -0.269123213}

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