8(4q-5)7q=20q-10

Solution for 8(4q-5)7q=20q-10 equation:

Simplifying
8(4q + -5) * 7q = 20q + -10

Reorder the terms:
8(-5 + 4q) * 7q = 20q + -10

Reorder the terms for easier multiplication:
8 * 7q(-5 + 4q) = 20q + -10

Multiply 8 * 7
56q(-5 + 4q) = 20q + -10
(-5 * 56q + 4q * 56q) = 20q + -10
(-280q + 224q2) = 20q + -10

Reorder the terms:
-280q + 224q2 = -10 + 20q

Solving
-280q + 224q2 = -10 + 20q

Solving for variable 'q'.

Reorder the terms:
10 + -280q + -20q + 224q2 = -10 + 20q + 10 + -20q

Combine like terms: -280q + -20q = -300q
10 + -300q + 224q2 = -10 + 20q + 10 + -20q

Reorder the terms:
10 + -300q + 224q2 = -10 + 10 + 20q + -20q

Combine like terms: -10 + 10 = 0
10 + -300q + 224q2 = 0 + 20q + -20q
10 + -300q + 224q2 = 20q + -20q

Combine like terms: 20q + -20q = 0
10 + -300q + 224q2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(5 + -150q + 112q2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(5 + -150q + 112q2)' equal to zero and attempt to solve:

Simplifying
5 + -150q + 112q2 = 0

Solving
5 + -150q + 112q2 = 0

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

Divide each side by '112'.
0.04464285714 + -1.339285714q + q2 = 0

Move the constant term to the right:

Add '-0.04464285714' to each side of the equation.
0.04464285714 + -1.339285714q + -0.04464285714 + q2 = 0 + -0.04464285714

Reorder the terms:
0.04464285714 + -0.04464285714 + -1.339285714q + q2 = 0 + -0.04464285714

Combine like terms: 0.04464285714 + -0.04464285714 = 0.00000000000
0.00000000000 + -1.339285714q + q2 = 0 + -0.04464285714
-1.339285714q + q2 = 0 + -0.04464285714

Combine like terms: 0 + -0.04464285714 = -0.04464285714
-1.339285714q + q2 = -0.04464285714

The q term is -1.339285714q.  Take half its coefficient (-0.669642857).
Square it (0.4484215559) and add it to both sides.

Add '0.4484215559' to each side of the equation.
-1.339285714q + 0.4484215559 + q2 = -0.04464285714 + 0.4484215559

Reorder the terms:
0.4484215559 + -1.339285714q + q2 = -0.04464285714 + 0.4484215559

Combine like terms: -0.04464285714 + 0.4484215559 = 0.40377869876
0.4484215559 + -1.339285714q + q2 = 0.40377869876

Factor a perfect square on the left side:
(q + -0.669642857)(q + -0.669642857) = 0.40377869876

Calculate the square root of the right side: 0.635435834

Break this problem into two subproblems by setting 
(q + -0.669642857) equal to 0.635435834 and -0.635435834.

Subproblem 1
q + -0.669642857 = 0.635435834

Simplifying
q + -0.669642857 = 0.635435834

Reorder the terms:
-0.669642857 + q = 0.635435834

Solving
-0.669642857 + q = 0.635435834

Solving for variable 'q'.

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

Add '0.669642857' to each side of the equation.
-0.669642857 + 0.669642857 + q = 0.635435834 + 0.669642857

Combine like terms: -0.669642857 + 0.669642857 = 0.000000000
0.000000000 + q = 0.635435834 + 0.669642857
q = 0.635435834 + 0.669642857

Combine like terms: 0.635435834 + 0.669642857 = 1.305078691
q = 1.305078691

Simplifying
q = 1.305078691

Subproblem 2
q + -0.669642857 = -0.635435834

Simplifying
q + -0.669642857 = -0.635435834

Reorder the terms:
-0.669642857 + q = -0.635435834

Solving
-0.669642857 + q = -0.635435834

Solving for variable 'q'.

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

Add '0.669642857' to each side of the equation.
-0.669642857 + 0.669642857 + q = -0.635435834 + 0.669642857

Combine like terms: -0.669642857 + 0.669642857 = 0.000000000
0.000000000 + q = -0.635435834 + 0.669642857
q = -0.635435834 + 0.669642857

Combine like terms: -0.635435834 + 0.669642857 = 0.034207023
q = 0.034207023

Simplifying
q = 0.034207023

Solution
The solution to the problem is based on the solutions
from the subproblems.
q = {1.305078691, 0.034207023}
Solution
q = {1.305078691, 0.034207023}