7y^2-16y-28=0

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Solution for 7y^2-16y-28=0 equation:


Simplifying
7y2 + -16y + -28 = 0

Reorder the terms:
-28 + -16y + 7y2 = 0

Solving
-28 + -16y + 7y2 = 0

Solving for variable 'y'.

Begin completing the square.  Divide all terms by
7 the coefficient of the squared term: 

Divide each side by '7'.
-4 + -2.285714286y + y2 = 0

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + -2.285714286y + 4 + y2 = 0 + 4

Reorder the terms:
-4 + 4 + -2.285714286y + y2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + -2.285714286y + y2 = 0 + 4
-2.285714286y + y2 = 0 + 4

Combine like terms: 0 + 4 = 4
-2.285714286y + y2 = 4

The y term is -2.285714286y.  Take half its coefficient (-1.142857143).
Square it (1.306122449) and add it to both sides.

Add '1.306122449' to each side of the equation.
-2.285714286y + 1.306122449 + y2 = 4 + 1.306122449

Reorder the terms:
1.306122449 + -2.285714286y + y2 = 4 + 1.306122449

Combine like terms: 4 + 1.306122449 = 5.306122449
1.306122449 + -2.285714286y + y2 = 5.306122449

Factor a perfect square on the left side:
(y + -1.142857143)(y + -1.142857143) = 5.306122449

Calculate the square root of the right side: 2.303502214

Break this problem into two subproblems by setting 
(y + -1.142857143) equal to 2.303502214 and -2.303502214.

Subproblem 1

y + -1.142857143 = 2.303502214 Simplifying y + -1.142857143 = 2.303502214 Reorder the terms: -1.142857143 + y = 2.303502214 Solving -1.142857143 + y = 2.303502214 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.142857143' to each side of the equation. -1.142857143 + 1.142857143 + y = 2.303502214 + 1.142857143 Combine like terms: -1.142857143 + 1.142857143 = 0.000000000 0.000000000 + y = 2.303502214 + 1.142857143 y = 2.303502214 + 1.142857143 Combine like terms: 2.303502214 + 1.142857143 = 3.446359357 y = 3.446359357 Simplifying y = 3.446359357

Subproblem 2

y + -1.142857143 = -2.303502214 Simplifying y + -1.142857143 = -2.303502214 Reorder the terms: -1.142857143 + y = -2.303502214 Solving -1.142857143 + y = -2.303502214 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.142857143' to each side of the equation. -1.142857143 + 1.142857143 + y = -2.303502214 + 1.142857143 Combine like terms: -1.142857143 + 1.142857143 = 0.000000000 0.000000000 + y = -2.303502214 + 1.142857143 y = -2.303502214 + 1.142857143 Combine like terms: -2.303502214 + 1.142857143 = -1.160645071 y = -1.160645071 Simplifying y = -1.160645071

Solution

The solution to the problem is based on the solutions from the subproblems. y = {3.446359357, -1.160645071}

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