-7a^2+32a+306=0

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Solution for -7a^2+32a+306=0 equation:


Simplifying
-7a2 + 32a + 306 = 0

Reorder the terms:
306 + 32a + -7a2 = 0

Solving
306 + 32a + -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'.
-43.71428571 + -4.571428571a + a2 = 0

Move the constant term to the right:

Add '43.71428571' to each side of the equation.
-43.71428571 + -4.571428571a + 43.71428571 + a2 = 0 + 43.71428571

Reorder the terms:
-43.71428571 + 43.71428571 + -4.571428571a + a2 = 0 + 43.71428571

Combine like terms: -43.71428571 + 43.71428571 = 0.00000000
0.00000000 + -4.571428571a + a2 = 0 + 43.71428571
-4.571428571a + a2 = 0 + 43.71428571

Combine like terms: 0 + 43.71428571 = 43.71428571
-4.571428571a + a2 = 43.71428571

The a term is -4.571428571a.  Take half its coefficient (-2.285714286).
Square it (5.224489797) and add it to both sides.

Add '5.224489797' to each side of the equation.
-4.571428571a + 5.224489797 + a2 = 43.71428571 + 5.224489797

Reorder the terms:
5.224489797 + -4.571428571a + a2 = 43.71428571 + 5.224489797

Combine like terms: 43.71428571 + 5.224489797 = 48.938775507
5.224489797 + -4.571428571a + a2 = 48.938775507

Factor a perfect square on the left side:
(a + -2.285714286)(a + -2.285714286) = 48.938775507

Calculate the square root of the right side: 6.995625455

Break this problem into two subproblems by setting 
(a + -2.285714286) equal to 6.995625455 and -6.995625455.

Subproblem 1

a + -2.285714286 = 6.995625455 Simplifying a + -2.285714286 = 6.995625455 Reorder the terms: -2.285714286 + a = 6.995625455 Solving -2.285714286 + a = 6.995625455 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2.285714286' to each side of the equation. -2.285714286 + 2.285714286 + a = 6.995625455 + 2.285714286 Combine like terms: -2.285714286 + 2.285714286 = 0.000000000 0.000000000 + a = 6.995625455 + 2.285714286 a = 6.995625455 + 2.285714286 Combine like terms: 6.995625455 + 2.285714286 = 9.281339741 a = 9.281339741 Simplifying a = 9.281339741

Subproblem 2

a + -2.285714286 = -6.995625455 Simplifying a + -2.285714286 = -6.995625455 Reorder the terms: -2.285714286 + a = -6.995625455 Solving -2.285714286 + a = -6.995625455 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2.285714286' to each side of the equation. -2.285714286 + 2.285714286 + a = -6.995625455 + 2.285714286 Combine like terms: -2.285714286 + 2.285714286 = 0.000000000 0.000000000 + a = -6.995625455 + 2.285714286 a = -6.995625455 + 2.285714286 Combine like terms: -6.995625455 + 2.285714286 = -4.709911169 a = -4.709911169 Simplifying a = -4.709911169

Solution

The solution to the problem is based on the solutions from the subproblems. a = {9.281339741, -4.709911169}

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