5x^2=-6x+1

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Solution for 5x^2=-6x+1 equation:


Simplifying
5x2 = -6x + 1

Reorder the terms:
5x2 = 1 + -6x

Solving
5x2 = 1 + -6x

Solving for variable 'x'.

Reorder the terms:
-1 + 6x + 5x2 = 1 + -6x + -1 + 6x

Reorder the terms:
-1 + 6x + 5x2 = 1 + -1 + -6x + 6x

Combine like terms: 1 + -1 = 0
-1 + 6x + 5x2 = 0 + -6x + 6x
-1 + 6x + 5x2 = -6x + 6x

Combine like terms: -6x + 6x = 0
-1 + 6x + 5x2 = 0

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

Divide each side by '5'.
-0.2 + 1.2x + x2 = 0

Move the constant term to the right:

Add '0.2' to each side of the equation.
-0.2 + 1.2x + 0.2 + x2 = 0 + 0.2

Reorder the terms:
-0.2 + 0.2 + 1.2x + x2 = 0 + 0.2

Combine like terms: -0.2 + 0.2 = 0.0
0.0 + 1.2x + x2 = 0 + 0.2
1.2x + x2 = 0 + 0.2

Combine like terms: 0 + 0.2 = 0.2
1.2x + x2 = 0.2

The x term is 1.2x.  Take half its coefficient (0.6).
Square it (0.36) and add it to both sides.

Add '0.36' to each side of the equation.
1.2x + 0.36 + x2 = 0.2 + 0.36

Reorder the terms:
0.36 + 1.2x + x2 = 0.2 + 0.36

Combine like terms: 0.2 + 0.36 = 0.56
0.36 + 1.2x + x2 = 0.56

Factor a perfect square on the left side:
(x + 0.6)(x + 0.6) = 0.56

Calculate the square root of the right side: 0.748331477

Break this problem into two subproblems by setting 
(x + 0.6) equal to 0.748331477 and -0.748331477.

Subproblem 1

x + 0.6 = 0.748331477 Simplifying x + 0.6 = 0.748331477 Reorder the terms: 0.6 + x = 0.748331477 Solving 0.6 + x = 0.748331477 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.6' to each side of the equation. 0.6 + -0.6 + x = 0.748331477 + -0.6 Combine like terms: 0.6 + -0.6 = 0.0 0.0 + x = 0.748331477 + -0.6 x = 0.748331477 + -0.6 Combine like terms: 0.748331477 + -0.6 = 0.148331477 x = 0.148331477 Simplifying x = 0.148331477

Subproblem 2

x + 0.6 = -0.748331477 Simplifying x + 0.6 = -0.748331477 Reorder the terms: 0.6 + x = -0.748331477 Solving 0.6 + x = -0.748331477 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.6' to each side of the equation. 0.6 + -0.6 + x = -0.748331477 + -0.6 Combine like terms: 0.6 + -0.6 = 0.0 0.0 + x = -0.748331477 + -0.6 x = -0.748331477 + -0.6 Combine like terms: -0.748331477 + -0.6 = -1.348331477 x = -1.348331477 Simplifying x = -1.348331477

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.148331477, -1.348331477}

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