3x^2-10x=5x^2-1

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Solution for 3x^2-10x=5x^2-1 equation:


Simplifying
3x2 + -10x = 5x2 + -1

Reorder the terms:
-10x + 3x2 = 5x2 + -1

Reorder the terms:
-10x + 3x2 = -1 + 5x2

Solving
-10x + 3x2 = -1 + 5x2

Solving for variable 'x'.

Reorder the terms:
1 + -10x + 3x2 + -5x2 = -1 + 5x2 + 1 + -5x2

Combine like terms: 3x2 + -5x2 = -2x2
1 + -10x + -2x2 = -1 + 5x2 + 1 + -5x2

Reorder the terms:
1 + -10x + -2x2 = -1 + 1 + 5x2 + -5x2

Combine like terms: -1 + 1 = 0
1 + -10x + -2x2 = 0 + 5x2 + -5x2
1 + -10x + -2x2 = 5x2 + -5x2

Combine like terms: 5x2 + -5x2 = 0
1 + -10x + -2x2 = 0

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

Divide each side by '-2'.
-0.5 + 5x + x2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + 5x + 0.5 + x2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + 5x + x2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + 5x + x2 = 0 + 0.5
5x + x2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
5x + x2 = 0.5

The x term is 5x.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5x + 6.25 + x2 = 0.5 + 6.25

Reorder the terms:
6.25 + 5x + x2 = 0.5 + 6.25

Combine like terms: 0.5 + 6.25 = 6.75
6.25 + 5x + x2 = 6.75

Factor a perfect square on the left side:
(x + 2.5)(x + 2.5) = 6.75

Calculate the square root of the right side: 2.598076211

Break this problem into two subproblems by setting 
(x + 2.5) equal to 2.598076211 and -2.598076211.

Subproblem 1

x + 2.5 = 2.598076211 Simplifying x + 2.5 = 2.598076211 Reorder the terms: 2.5 + x = 2.598076211 Solving 2.5 + x = 2.598076211 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + x = 2.598076211 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = 2.598076211 + -2.5 x = 2.598076211 + -2.5 Combine like terms: 2.598076211 + -2.5 = 0.098076211 x = 0.098076211 Simplifying x = 0.098076211

Subproblem 2

x + 2.5 = -2.598076211 Simplifying x + 2.5 = -2.598076211 Reorder the terms: 2.5 + x = -2.598076211 Solving 2.5 + x = -2.598076211 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + x = -2.598076211 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = -2.598076211 + -2.5 x = -2.598076211 + -2.5 Combine like terms: -2.598076211 + -2.5 = -5.098076211 x = -5.098076211 Simplifying x = -5.098076211

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.098076211, -5.098076211}

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