6y^2+7y-15=10

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Solution for 6y^2+7y-15=10 equation:


Simplifying
6y2 + 7y + -15 = 10

Reorder the terms:
-15 + 7y + 6y2 = 10

Solving
-15 + 7y + 6y2 = 10

Solving for variable 'y'.

Reorder the terms:
-15 + -10 + 7y + 6y2 = 10 + -10

Combine like terms: -15 + -10 = -25
-25 + 7y + 6y2 = 10 + -10

Combine like terms: 10 + -10 = 0
-25 + 7y + 6y2 = 0

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

Divide each side by '6'.
-4.166666667 + 1.166666667y + y2 = 0

Move the constant term to the right:

Add '4.166666667' to each side of the equation.
-4.166666667 + 1.166666667y + 4.166666667 + y2 = 0 + 4.166666667

Reorder the terms:
-4.166666667 + 4.166666667 + 1.166666667y + y2 = 0 + 4.166666667

Combine like terms: -4.166666667 + 4.166666667 = 0.000000000
0.000000000 + 1.166666667y + y2 = 0 + 4.166666667
1.166666667y + y2 = 0 + 4.166666667

Combine like terms: 0 + 4.166666667 = 4.166666667
1.166666667y + y2 = 4.166666667

The y term is 1.166666667y.  Take half its coefficient (0.5833333335).
Square it (0.3402777780) and add it to both sides.

Add '0.3402777780' to each side of the equation.
1.166666667y + 0.3402777780 + y2 = 4.166666667 + 0.3402777780

Reorder the terms:
0.3402777780 + 1.166666667y + y2 = 4.166666667 + 0.3402777780

Combine like terms: 4.166666667 + 0.3402777780 = 4.506944445
0.3402777780 + 1.166666667y + y2 = 4.506944445

Factor a perfect square on the left side:
(y + 0.5833333335)(y + 0.5833333335) = 4.506944445

Calculate the square root of the right side: 2.122956534

Break this problem into two subproblems by setting 
(y + 0.5833333335) equal to 2.122956534 and -2.122956534.

Subproblem 1

y + 0.5833333335 = 2.122956534 Simplifying y + 0.5833333335 = 2.122956534 Reorder the terms: 0.5833333335 + y = 2.122956534 Solving 0.5833333335 + y = 2.122956534 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.5833333335' to each side of the equation. 0.5833333335 + -0.5833333335 + y = 2.122956534 + -0.5833333335 Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000 0.0000000000 + y = 2.122956534 + -0.5833333335 y = 2.122956534 + -0.5833333335 Combine like terms: 2.122956534 + -0.5833333335 = 1.5396232005 y = 1.5396232005 Simplifying y = 1.5396232005

Subproblem 2

y + 0.5833333335 = -2.122956534 Simplifying y + 0.5833333335 = -2.122956534 Reorder the terms: 0.5833333335 + y = -2.122956534 Solving 0.5833333335 + y = -2.122956534 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.5833333335' to each side of the equation. 0.5833333335 + -0.5833333335 + y = -2.122956534 + -0.5833333335 Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000 0.0000000000 + y = -2.122956534 + -0.5833333335 y = -2.122956534 + -0.5833333335 Combine like terms: -2.122956534 + -0.5833333335 = -2.7062898675 y = -2.7062898675 Simplifying y = -2.7062898675

Solution

The solution to the problem is based on the solutions from the subproblems. y = {1.5396232005, -2.7062898675}

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