0.25q^2-6q+40=0.5q^2-7q+35

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Solution for 0.25q^2-6q+40=0.5q^2-7q+35 equation:


Simplifying
0.25q2 + -6q + 40 = 0.5q2 + -7q + 35

Reorder the terms:
40 + -6q + 0.25q2 = 0.5q2 + -7q + 35

Reorder the terms:
40 + -6q + 0.25q2 = 35 + -7q + 0.5q2

Solving
40 + -6q + 0.25q2 = 35 + -7q + 0.5q2

Solving for variable 'q'.

Reorder the terms:
40 + -35 + -6q + 7q + 0.25q2 + -0.5q2 = 35 + -7q + 0.5q2 + -35 + 7q + -0.5q2

Combine like terms: 40 + -35 = 5
5 + -6q + 7q + 0.25q2 + -0.5q2 = 35 + -7q + 0.5q2 + -35 + 7q + -0.5q2

Combine like terms: -6q + 7q = 1q
5 + 1q + 0.25q2 + -0.5q2 = 35 + -7q + 0.5q2 + -35 + 7q + -0.5q2

Combine like terms: 0.25q2 + -0.5q2 = -0.25q2
5 + 1q + -0.25q2 = 35 + -7q + 0.5q2 + -35 + 7q + -0.5q2

Reorder the terms:
5 + 1q + -0.25q2 = 35 + -35 + -7q + 7q + 0.5q2 + -0.5q2

Combine like terms: 35 + -35 = 0
5 + 1q + -0.25q2 = 0 + -7q + 7q + 0.5q2 + -0.5q2
5 + 1q + -0.25q2 = -7q + 7q + 0.5q2 + -0.5q2

Combine like terms: -7q + 7q = 0
5 + 1q + -0.25q2 = 0 + 0.5q2 + -0.5q2
5 + 1q + -0.25q2 = 0.5q2 + -0.5q2

Combine like terms: 0.5q2 + -0.5q2 = 0.0
5 + 1q + -0.25q2 = 0.0

Begin completing the square.  Divide all terms by
-0.25 the coefficient of the squared term: 

Divide each side by '-0.25'.
-20 + -4q + q2 = 0

Move the constant term to the right:

Add '20' to each side of the equation.
-20 + -4q + 20 + q2 = 0 + 20

Reorder the terms:
-20 + 20 + -4q + q2 = 0 + 20

Combine like terms: -20 + 20 = 0
0 + -4q + q2 = 0 + 20
-4q + q2 = 0 + 20

Combine like terms: 0 + 20 = 20
-4q + q2 = 20

The q term is -4q.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
-4q + 4 + q2 = 20 + 4

Reorder the terms:
4 + -4q + q2 = 20 + 4

Combine like terms: 20 + 4 = 24
4 + -4q + q2 = 24

Factor a perfect square on the left side:
(q + -2)(q + -2) = 24

Calculate the square root of the right side: 4.898979486

Break this problem into two subproblems by setting 
(q + -2) equal to 4.898979486 and -4.898979486.

Subproblem 1

q + -2 = 4.898979486 Simplifying q + -2 = 4.898979486 Reorder the terms: -2 + q = 4.898979486 Solving -2 + q = 4.898979486 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + q = 4.898979486 + 2 Combine like terms: -2 + 2 = 0 0 + q = 4.898979486 + 2 q = 4.898979486 + 2 Combine like terms: 4.898979486 + 2 = 6.898979486 q = 6.898979486 Simplifying q = 6.898979486

Subproblem 2

q + -2 = -4.898979486 Simplifying q + -2 = -4.898979486 Reorder the terms: -2 + q = -4.898979486 Solving -2 + q = -4.898979486 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + q = -4.898979486 + 2 Combine like terms: -2 + 2 = 0 0 + q = -4.898979486 + 2 q = -4.898979486 + 2 Combine like terms: -4.898979486 + 2 = -2.898979486 q = -2.898979486 Simplifying q = -2.898979486

Solution

The solution to the problem is based on the solutions from the subproblems. q = {6.898979486, -2.898979486}

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