10b^2+3b=75

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Solution for 10b^2+3b=75 equation:


Simplifying
10b2 + 3b = 75

Reorder the terms:
3b + 10b2 = 75

Solving
3b + 10b2 = 75

Solving for variable 'b'.

Reorder the terms:
-75 + 3b + 10b2 = 75 + -75

Combine like terms: 75 + -75 = 0
-75 + 3b + 10b2 = 0

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

Divide each side by '10'.
-7.5 + 0.3b + b2 = 0

Move the constant term to the right:

Add '7.5' to each side of the equation.
-7.5 + 0.3b + 7.5 + b2 = 0 + 7.5

Reorder the terms:
-7.5 + 7.5 + 0.3b + b2 = 0 + 7.5

Combine like terms: -7.5 + 7.5 = 0.0
0.0 + 0.3b + b2 = 0 + 7.5
0.3b + b2 = 0 + 7.5

Combine like terms: 0 + 7.5 = 7.5
0.3b + b2 = 7.5

The b term is 0.3b.  Take half its coefficient (0.15).
Square it (0.0225) and add it to both sides.

Add '0.0225' to each side of the equation.
0.3b + 0.0225 + b2 = 7.5 + 0.0225

Reorder the terms:
0.0225 + 0.3b + b2 = 7.5 + 0.0225

Combine like terms: 7.5 + 0.0225 = 7.5225
0.0225 + 0.3b + b2 = 7.5225

Factor a perfect square on the left side:
(b + 0.15)(b + 0.15) = 7.5225

Calculate the square root of the right side: 2.74271763

Break this problem into two subproblems by setting 
(b + 0.15) equal to 2.74271763 and -2.74271763.

Subproblem 1

b + 0.15 = 2.74271763 Simplifying b + 0.15 = 2.74271763 Reorder the terms: 0.15 + b = 2.74271763 Solving 0.15 + b = 2.74271763 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.15' to each side of the equation. 0.15 + -0.15 + b = 2.74271763 + -0.15 Combine like terms: 0.15 + -0.15 = 0.00 0.00 + b = 2.74271763 + -0.15 b = 2.74271763 + -0.15 Combine like terms: 2.74271763 + -0.15 = 2.59271763 b = 2.59271763 Simplifying b = 2.59271763

Subproblem 2

b + 0.15 = -2.74271763 Simplifying b + 0.15 = -2.74271763 Reorder the terms: 0.15 + b = -2.74271763 Solving 0.15 + b = -2.74271763 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.15' to each side of the equation. 0.15 + -0.15 + b = -2.74271763 + -0.15 Combine like terms: 0.15 + -0.15 = 0.00 0.00 + b = -2.74271763 + -0.15 b = -2.74271763 + -0.15 Combine like terms: -2.74271763 + -0.15 = -2.89271763 b = -2.89271763 Simplifying b = -2.89271763

Solution

The solution to the problem is based on the solutions from the subproblems. b = {2.59271763, -2.89271763}

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