4q+7=q^2-5q+1

Solution for 4q+7=q^2-5q+1 equation:

Simplifying
4q + 7 = q2 + -5q + 1

Reorder the terms:
7 + 4q = q2 + -5q + 1

Reorder the terms:
7 + 4q = 1 + -5q + q2

Solving
7 + 4q = 1 + -5q + q2

Solving for variable 'q'.

Reorder the terms:
7 + -1 + 4q + 5q + -1q2 = 1 + -5q + q2 + -1 + 5q + -1q2

Combine like terms: 7 + -1 = 6
6 + 4q + 5q + -1q2 = 1 + -5q + q2 + -1 + 5q + -1q2

Combine like terms: 4q + 5q = 9q
6 + 9q + -1q2 = 1 + -5q + q2 + -1 + 5q + -1q2

Reorder the terms:
6 + 9q + -1q2 = 1 + -1 + -5q + 5q + q2 + -1q2

Combine like terms: 1 + -1 = 0
6 + 9q + -1q2 = 0 + -5q + 5q + q2 + -1q2
6 + 9q + -1q2 = -5q + 5q + q2 + -1q2

Combine like terms: -5q + 5q = 0
6 + 9q + -1q2 = 0 + q2 + -1q2
6 + 9q + -1q2 = q2 + -1q2

Combine like terms: q2 + -1q2 = 0
6 + 9q + -1q2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-6 + -9q + q2 = 0

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + -9q + 6 + q2 = 0 + 6

Reorder the terms:
-6 + 6 + -9q + q2 = 0 + 6

Combine like terms: -6 + 6 = 0
0 + -9q + q2 = 0 + 6
-9q + q2 = 0 + 6

Combine like terms: 0 + 6 = 6
-9q + q2 = 6

The q term is -9q.  Take half its coefficient (-4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
-9q + 20.25 + q2 = 6 + 20.25

Reorder the terms:
20.25 + -9q + q2 = 6 + 20.25

Combine like terms: 6 + 20.25 = 26.25
20.25 + -9q + q2 = 26.25

Factor a perfect square on the left side:
(q + -4.5)(q + -4.5) = 26.25

Calculate the square root of the right side: 5.123475383

Break this problem into two subproblems by setting 
(q + -4.5) equal to 5.123475383 and -5.123475383.

Subproblem 1
q + -4.5 = 5.123475383

Simplifying
q + -4.5 = 5.123475383

Reorder the terms:
-4.5 + q = 5.123475383

Solving
-4.5 + q = 5.123475383

Solving for variable 'q'.

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

Add '4.5' to each side of the equation.
-4.5 + 4.5 + q = 5.123475383 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + q = 5.123475383 + 4.5
q = 5.123475383 + 4.5

Combine like terms: 5.123475383 + 4.5 = 9.623475383
q = 9.623475383

Simplifying
q = 9.623475383

Subproblem 2
q + -4.5 = -5.123475383

Simplifying
q + -4.5 = -5.123475383

Reorder the terms:
-4.5 + q = -5.123475383

Solving
-4.5 + q = -5.123475383

Solving for variable 'q'.

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

Add '4.5' to each side of the equation.
-4.5 + 4.5 + q = -5.123475383 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + q = -5.123475383 + 4.5
q = -5.123475383 + 4.5

Combine like terms: -5.123475383 + 4.5 = -0.623475383
q = -0.623475383

Simplifying
q = -0.623475383

Solution
The solution to the problem is based on the solutions
from the subproblems.
q = {9.623475383, -0.623475383}