3x(4x-5)=10x^2+75

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Solution for 3x(4x-5)=10x^2+75 equation:


Simplifying
3x(4x + -5) = 10x2 + 75

Reorder the terms:
3x(-5 + 4x) = 10x2 + 75
(-5 * 3x + 4x * 3x) = 10x2 + 75
(-15x + 12x2) = 10x2 + 75

Reorder the terms:
-15x + 12x2 = 75 + 10x2

Solving
-15x + 12x2 = 75 + 10x2

Solving for variable 'x'.

Reorder the terms:
-75 + -15x + 12x2 + -10x2 = 75 + 10x2 + -75 + -10x2

Combine like terms: 12x2 + -10x2 = 2x2
-75 + -15x + 2x2 = 75 + 10x2 + -75 + -10x2

Reorder the terms:
-75 + -15x + 2x2 = 75 + -75 + 10x2 + -10x2

Combine like terms: 75 + -75 = 0
-75 + -15x + 2x2 = 0 + 10x2 + -10x2
-75 + -15x + 2x2 = 10x2 + -10x2

Combine like terms: 10x2 + -10x2 = 0
-75 + -15x + 2x2 = 0

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

Divide each side by '2'.
-37.5 + -7.5x + x2 = 0

Move the constant term to the right:

Add '37.5' to each side of the equation.
-37.5 + -7.5x + 37.5 + x2 = 0 + 37.5

Reorder the terms:
-37.5 + 37.5 + -7.5x + x2 = 0 + 37.5

Combine like terms: -37.5 + 37.5 = 0.0
0.0 + -7.5x + x2 = 0 + 37.5
-7.5x + x2 = 0 + 37.5

Combine like terms: 0 + 37.5 = 37.5
-7.5x + x2 = 37.5

The x term is -7.5x.  Take half its coefficient (-3.75).
Square it (14.0625) and add it to both sides.

Add '14.0625' to each side of the equation.
-7.5x + 14.0625 + x2 = 37.5 + 14.0625

Reorder the terms:
14.0625 + -7.5x + x2 = 37.5 + 14.0625

Combine like terms: 37.5 + 14.0625 = 51.5625
14.0625 + -7.5x + x2 = 51.5625

Factor a perfect square on the left side:
(x + -3.75)(x + -3.75) = 51.5625

Calculate the square root of the right side: 7.180703308

Break this problem into two subproblems by setting 
(x + -3.75) equal to 7.180703308 and -7.180703308.

Subproblem 1

x + -3.75 = 7.180703308 Simplifying x + -3.75 = 7.180703308 Reorder the terms: -3.75 + x = 7.180703308 Solving -3.75 + x = 7.180703308 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.75' to each side of the equation. -3.75 + 3.75 + x = 7.180703308 + 3.75 Combine like terms: -3.75 + 3.75 = 0.00 0.00 + x = 7.180703308 + 3.75 x = 7.180703308 + 3.75 Combine like terms: 7.180703308 + 3.75 = 10.930703308 x = 10.930703308 Simplifying x = 10.930703308

Subproblem 2

x + -3.75 = -7.180703308 Simplifying x + -3.75 = -7.180703308 Reorder the terms: -3.75 + x = -7.180703308 Solving -3.75 + x = -7.180703308 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.75' to each side of the equation. -3.75 + 3.75 + x = -7.180703308 + 3.75 Combine like terms: -3.75 + 3.75 = 0.00 0.00 + x = -7.180703308 + 3.75 x = -7.180703308 + 3.75 Combine like terms: -7.180703308 + 3.75 = -3.430703308 x = -3.430703308 Simplifying x = -3.430703308

Solution

The solution to the problem is based on the solutions from the subproblems. x = {10.930703308, -3.430703308}

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