7y^2-4y-6=0

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Solution for 7y^2-4y-6=0 equation:


Simplifying
7y2 + -4y + -6 = 0

Reorder the terms:
-6 + -4y + 7y2 = 0

Solving
-6 + -4y + 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'.
-0.8571428571 + -0.5714285714y + y2 = 0

Move the constant term to the right:

Add '0.8571428571' to each side of the equation.
-0.8571428571 + -0.5714285714y + 0.8571428571 + y2 = 0 + 0.8571428571

Reorder the terms:
-0.8571428571 + 0.8571428571 + -0.5714285714y + y2 = 0 + 0.8571428571

Combine like terms: -0.8571428571 + 0.8571428571 = 0.0000000000
0.0000000000 + -0.5714285714y + y2 = 0 + 0.8571428571
-0.5714285714y + y2 = 0 + 0.8571428571

Combine like terms: 0 + 0.8571428571 = 0.8571428571
-0.5714285714y + y2 = 0.8571428571

The y term is -0.5714285714y.  Take half its coefficient (-0.2857142857).
Square it (0.08163265305) and add it to both sides.

Add '0.08163265305' to each side of the equation.
-0.5714285714y + 0.08163265305 + y2 = 0.8571428571 + 0.08163265305

Reorder the terms:
0.08163265305 + -0.5714285714y + y2 = 0.8571428571 + 0.08163265305

Combine like terms: 0.8571428571 + 0.08163265305 = 0.93877551015
0.08163265305 + -0.5714285714y + y2 = 0.93877551015

Factor a perfect square on the left side:
(y + -0.2857142857)(y + -0.2857142857) = 0.93877551015

Calculate the square root of the right side: 0.968904283

Break this problem into two subproblems by setting 
(y + -0.2857142857) equal to 0.968904283 and -0.968904283.

Subproblem 1

y + -0.2857142857 = 0.968904283 Simplifying y + -0.2857142857 = 0.968904283 Reorder the terms: -0.2857142857 + y = 0.968904283 Solving -0.2857142857 + y = 0.968904283 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.2857142857' to each side of the equation. -0.2857142857 + 0.2857142857 + y = 0.968904283 + 0.2857142857 Combine like terms: -0.2857142857 + 0.2857142857 = 0.0000000000 0.0000000000 + y = 0.968904283 + 0.2857142857 y = 0.968904283 + 0.2857142857 Combine like terms: 0.968904283 + 0.2857142857 = 1.2546185687 y = 1.2546185687 Simplifying y = 1.2546185687

Subproblem 2

y + -0.2857142857 = -0.968904283 Simplifying y + -0.2857142857 = -0.968904283 Reorder the terms: -0.2857142857 + y = -0.968904283 Solving -0.2857142857 + y = -0.968904283 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.2857142857' to each side of the equation. -0.2857142857 + 0.2857142857 + y = -0.968904283 + 0.2857142857 Combine like terms: -0.2857142857 + 0.2857142857 = 0.0000000000 0.0000000000 + y = -0.968904283 + 0.2857142857 y = -0.968904283 + 0.2857142857 Combine like terms: -0.968904283 + 0.2857142857 = -0.6831899973 y = -0.6831899973 Simplifying y = -0.6831899973

Solution

The solution to the problem is based on the solutions from the subproblems. y = {1.2546185687, -0.6831899973}

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