-4y^2-7y=1

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Solution for -4y^2-7y=1 equation: 


Simplifying
-4y2 + -7y = 1

Reorder the terms:
-7y + -4y2 = 1

Solving
-7y + -4y2 = 1

Solving for variable 'y'.

Reorder the terms:
-1 + -7y + -4y2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + -7y + -4y2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(1 + 7y + 4y2) = 0

Ignore the factor -1.

Subproblem 1
Set the factor '(1 + 7y + 4y2)' equal to zero and attempt to solve:

Simplifying
1 + 7y + 4y2 = 0

Solving
1 + 7y + 4y2 = 0

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

Divide each side by '4'.
0.25 + 1.75y + y2 = 0

Move the constant term to the right:

Add '-0.25' to each side of the equation.
0.25 + 1.75y + -0.25 + y2 = 0 + -0.25

Reorder the terms:
0.25 + -0.25 + 1.75y + y2 = 0 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + 1.75y + y2 = 0 + -0.25
1.75y + y2 = 0 + -0.25

Combine like terms: 0 + -0.25 = -0.25
1.75y + y2 = -0.25

The y term is 1.75y.  Take half its coefficient (0.875).
Square it (0.765625) and add it to both sides.

Add '0.765625' to each side of the equation.
1.75y + 0.765625 + y2 = -0.25 + 0.765625

Reorder the terms:
0.765625 + 1.75y + y2 = -0.25 + 0.765625

Combine like terms: -0.25 + 0.765625 = 0.515625
0.765625 + 1.75y + y2 = 0.515625

Factor a perfect square on the left side:
(y + 0.875)(y + 0.875) = 0.515625

Calculate the square root of the right side: 0.718070331

Break this problem into two subproblems by setting 
(y + 0.875) equal to 0.718070331 and -0.718070331.


Subproblem 1
y + 0.875 = 0.718070331

Simplifying
y + 0.875 = 0.718070331

Reorder the terms:
0.875 + y = 0.718070331

Solving
0.875 + y = 0.718070331

Solving for variable 'y'.

Move all terms containing y to the left, all other terms to the right.

Add '-0.875' to each side of the equation.
0.875 + -0.875 + y = 0.718070331 + -0.875

Combine like terms: 0.875 + -0.875 = 0.000
0.000 + y = 0.718070331 + -0.875
y = 0.718070331 + -0.875

Combine like terms: 0.718070331 + -0.875 = -0.156929669
y = -0.156929669

Simplifying
y = -0.156929669


Subproblem 2
y + 0.875 = -0.718070331

Simplifying
y + 0.875 = -0.718070331

Reorder the terms:
0.875 + y = -0.718070331

Solving
0.875 + y = -0.718070331

Solving for variable 'y'.

Move all terms containing y to the left, all other terms to the right.

Add '-0.875' to each side of the equation.
0.875 + -0.875 + y = -0.718070331 + -0.875

Combine like terms: 0.875 + -0.875 = 0.000
0.000 + y = -0.718070331 + -0.875
y = -0.718070331 + -0.875

Combine like terms: -0.718070331 + -0.875 = -1.593070331
y = -1.593070331

Simplifying
y = -1.593070331


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {-0.156929669, -1.593070331}
Solution
y = {-0.156929669, -1.593070331}

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