10x^2+30x=2x^2-30x-75

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Solution for 10x^2+30x=2x^2-30x-75 equation:


Simplifying
10x2 + 30x = 2x2 + -30x + -75

Reorder the terms:
30x + 10x2 = 2x2 + -30x + -75

Reorder the terms:
30x + 10x2 = -75 + -30x + 2x2

Solving
30x + 10x2 = -75 + -30x + 2x2

Solving for variable 'x'.

Reorder the terms:
75 + 30x + 30x + 10x2 + -2x2 = -75 + -30x + 2x2 + 75 + 30x + -2x2

Combine like terms: 30x + 30x = 60x
75 + 60x + 10x2 + -2x2 = -75 + -30x + 2x2 + 75 + 30x + -2x2

Combine like terms: 10x2 + -2x2 = 8x2
75 + 60x + 8x2 = -75 + -30x + 2x2 + 75 + 30x + -2x2

Reorder the terms:
75 + 60x + 8x2 = -75 + 75 + -30x + 30x + 2x2 + -2x2

Combine like terms: -75 + 75 = 0
75 + 60x + 8x2 = 0 + -30x + 30x + 2x2 + -2x2
75 + 60x + 8x2 = -30x + 30x + 2x2 + -2x2

Combine like terms: -30x + 30x = 0
75 + 60x + 8x2 = 0 + 2x2 + -2x2
75 + 60x + 8x2 = 2x2 + -2x2

Combine like terms: 2x2 + -2x2 = 0
75 + 60x + 8x2 = 0

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

Divide each side by '8'.
9.375 + 7.5x + x2 = 0

Move the constant term to the right:

Add '-9.375' to each side of the equation.
9.375 + 7.5x + -9.375 + x2 = 0 + -9.375

Reorder the terms:
9.375 + -9.375 + 7.5x + x2 = 0 + -9.375

Combine like terms: 9.375 + -9.375 = 0.000
0.000 + 7.5x + x2 = 0 + -9.375
7.5x + x2 = 0 + -9.375

Combine like terms: 0 + -9.375 = -9.375
7.5x + x2 = -9.375

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 = -9.375 + 14.0625

Reorder the terms:
14.0625 + 7.5x + x2 = -9.375 + 14.0625

Combine like terms: -9.375 + 14.0625 = 4.6875
14.0625 + 7.5x + x2 = 4.6875

Factor a perfect square on the left side:
(x + 3.75)(x + 3.75) = 4.6875

Calculate the square root of the right side: 2.165063509

Break this problem into two subproblems by setting 
(x + 3.75) equal to 2.165063509 and -2.165063509.

Subproblem 1

x + 3.75 = 2.165063509 Simplifying x + 3.75 = 2.165063509 Reorder the terms: 3.75 + x = 2.165063509 Solving 3.75 + x = 2.165063509 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 = 2.165063509 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + x = 2.165063509 + -3.75 x = 2.165063509 + -3.75 Combine like terms: 2.165063509 + -3.75 = -1.584936491 x = -1.584936491 Simplifying x = -1.584936491

Subproblem 2

x + 3.75 = -2.165063509 Simplifying x + 3.75 = -2.165063509 Reorder the terms: 3.75 + x = -2.165063509 Solving 3.75 + x = -2.165063509 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 = -2.165063509 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + x = -2.165063509 + -3.75 x = -2.165063509 + -3.75 Combine like terms: -2.165063509 + -3.75 = -5.915063509 x = -5.915063509 Simplifying x = -5.915063509

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-1.584936491, -5.915063509}

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