10x^2=2x+1

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Solution for 10x^2=2x+1 equation:


Simplifying
10x2 = 2x + 1

Reorder the terms:
10x2 = 1 + 2x

Solving
10x2 = 1 + 2x

Solving for variable 'x'.

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

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

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

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

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

Divide each side by '10'.
-0.1 + -0.2x + x2 = 0

Move the constant term to the right:

Add '0.1' to each side of the equation.
-0.1 + -0.2x + 0.1 + x2 = 0 + 0.1

Reorder the terms:
-0.1 + 0.1 + -0.2x + x2 = 0 + 0.1

Combine like terms: -0.1 + 0.1 = 0.0
0.0 + -0.2x + x2 = 0 + 0.1
-0.2x + x2 = 0 + 0.1

Combine like terms: 0 + 0.1 = 0.1
-0.2x + x2 = 0.1

The x term is -0.2x.  Take half its coefficient (-0.1).
Square it (0.01) and add it to both sides.

Add '0.01' to each side of the equation.
-0.2x + 0.01 + x2 = 0.1 + 0.01

Reorder the terms:
0.01 + -0.2x + x2 = 0.1 + 0.01

Combine like terms: 0.1 + 0.01 = 0.11
0.01 + -0.2x + x2 = 0.11

Factor a perfect square on the left side:
(x + -0.1)(x + -0.1) = 0.11

Calculate the square root of the right side: 0.331662479

Break this problem into two subproblems by setting 
(x + -0.1) equal to 0.331662479 and -0.331662479.

Subproblem 1

x + -0.1 = 0.331662479 Simplifying x + -0.1 = 0.331662479 Reorder the terms: -0.1 + x = 0.331662479 Solving -0.1 + x = 0.331662479 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1' to each side of the equation. -0.1 + 0.1 + x = 0.331662479 + 0.1 Combine like terms: -0.1 + 0.1 = 0.0 0.0 + x = 0.331662479 + 0.1 x = 0.331662479 + 0.1 Combine like terms: 0.331662479 + 0.1 = 0.431662479 x = 0.431662479 Simplifying x = 0.431662479

Subproblem 2

x + -0.1 = -0.331662479 Simplifying x + -0.1 = -0.331662479 Reorder the terms: -0.1 + x = -0.331662479 Solving -0.1 + x = -0.331662479 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1' to each side of the equation. -0.1 + 0.1 + x = -0.331662479 + 0.1 Combine like terms: -0.1 + 0.1 = 0.0 0.0 + x = -0.331662479 + 0.1 x = -0.331662479 + 0.1 Combine like terms: -0.331662479 + 0.1 = -0.231662479 x = -0.231662479 Simplifying x = -0.231662479

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.431662479, -0.231662479}

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