5x^2-2x=1

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


Simplifying
5x2 + -2x = 1

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

Solving
-2x + 5x2 = 1

Solving for variable 'x'.

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

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

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

Divide each side by '5'.
-0.2 + -0.4x + x2 = 0

Move the constant term to the right:

Add '0.2' to each side of the equation.
-0.2 + -0.4x + 0.2 + x2 = 0 + 0.2

Reorder the terms:
-0.2 + 0.2 + -0.4x + x2 = 0 + 0.2

Combine like terms: -0.2 + 0.2 = 0.0
0.0 + -0.4x + x2 = 0 + 0.2
-0.4x + x2 = 0 + 0.2

Combine like terms: 0 + 0.2 = 0.2
-0.4x + x2 = 0.2

The x term is -0.4x.  Take half its coefficient (-0.2).
Square it (0.04) and add it to both sides.

Add '0.04' to each side of the equation.
-0.4x + 0.04 + x2 = 0.2 + 0.04

Reorder the terms:
0.04 + -0.4x + x2 = 0.2 + 0.04

Combine like terms: 0.2 + 0.04 = 0.24
0.04 + -0.4x + x2 = 0.24

Factor a perfect square on the left side:
(x + -0.2)(x + -0.2) = 0.24

Calculate the square root of the right side: 0.489897949

Break this problem into two subproblems by setting 
(x + -0.2) equal to 0.489897949 and -0.489897949.

Subproblem 1

x + -0.2 = 0.489897949 Simplifying x + -0.2 = 0.489897949 Reorder the terms: -0.2 + x = 0.489897949 Solving -0.2 + x = 0.489897949 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.2' to each side of the equation. -0.2 + 0.2 + x = 0.489897949 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + x = 0.489897949 + 0.2 x = 0.489897949 + 0.2 Combine like terms: 0.489897949 + 0.2 = 0.689897949 x = 0.689897949 Simplifying x = 0.689897949

Subproblem 2

x + -0.2 = -0.489897949 Simplifying x + -0.2 = -0.489897949 Reorder the terms: -0.2 + x = -0.489897949 Solving -0.2 + x = -0.489897949 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.2' to each side of the equation. -0.2 + 0.2 + x = -0.489897949 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + x = -0.489897949 + 0.2 x = -0.489897949 + 0.2 Combine like terms: -0.489897949 + 0.2 = -0.289897949 x = -0.289897949 Simplifying x = -0.289897949

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.689897949, -0.289897949}

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