10x=5x^2-1

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


Simplifying
10x = 5x2 + -1

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

Solving
10x = -1 + 5x2

Solving for variable 'x'.

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

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

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

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

Begin completing the square.  Divide all terms by
-5 the coefficient of the squared term: 

Divide each side by '-5'.
-0.2 + -2x + x2 = 0

Move the constant term to the right:

Add '0.2' to each side of the equation.
-0.2 + -2x + 0.2 + x2 = 0 + 0.2

Reorder the terms:
-0.2 + 0.2 + -2x + x2 = 0 + 0.2

Combine like terms: -0.2 + 0.2 = 0.0
0.0 + -2x + x2 = 0 + 0.2
-2x + x2 = 0 + 0.2

Combine like terms: 0 + 0.2 = 0.2
-2x + x2 = 0.2

The x term is -2x.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2x + 1 + x2 = 0.2 + 1

Reorder the terms:
1 + -2x + x2 = 0.2 + 1

Combine like terms: 0.2 + 1 = 1.2
1 + -2x + x2 = 1.2

Factor a perfect square on the left side:
(x + -1)(x + -1) = 1.2

Calculate the square root of the right side: 1.095445115

Break this problem into two subproblems by setting 
(x + -1) equal to 1.095445115 and -1.095445115.

Subproblem 1

x + -1 = 1.095445115 Simplifying x + -1 = 1.095445115 Reorder the terms: -1 + x = 1.095445115 Solving -1 + x = 1.095445115 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = 1.095445115 + 1 Combine like terms: -1 + 1 = 0 0 + x = 1.095445115 + 1 x = 1.095445115 + 1 Combine like terms: 1.095445115 + 1 = 2.095445115 x = 2.095445115 Simplifying x = 2.095445115

Subproblem 2

x + -1 = -1.095445115 Simplifying x + -1 = -1.095445115 Reorder the terms: -1 + x = -1.095445115 Solving -1 + x = -1.095445115 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = -1.095445115 + 1 Combine like terms: -1 + 1 = 0 0 + x = -1.095445115 + 1 x = -1.095445115 + 1 Combine like terms: -1.095445115 + 1 = -0.095445115 x = -0.095445115 Simplifying x = -0.095445115

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.095445115, -0.095445115}

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