5x^2+3x=1

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Solution for 5x^2+3x=1 equation:


Simplifying
5x2 + 3x = 1

Reorder the terms:
3x + 5x2 = 1

Solving
3x + 5x2 = 1

Solving for variable 'x'.

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

Combine like terms: 1 + -1 = 0
-1 + 3x + 5x2 = 0

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

Divide each side by '5'.
-0.2 + 0.6x + x2 = 0

Move the constant term to the right:

Add '0.2' to each side of the equation.
-0.2 + 0.6x + 0.2 + x2 = 0 + 0.2

Reorder the terms:
-0.2 + 0.2 + 0.6x + x2 = 0 + 0.2

Combine like terms: -0.2 + 0.2 = 0.0
0.0 + 0.6x + x2 = 0 + 0.2
0.6x + x2 = 0 + 0.2

Combine like terms: 0 + 0.2 = 0.2
0.6x + x2 = 0.2

The x term is 0.6x.  Take half its coefficient (0.3).
Square it (0.09) and add it to both sides.

Add '0.09' to each side of the equation.
0.6x + 0.09 + x2 = 0.2 + 0.09

Reorder the terms:
0.09 + 0.6x + x2 = 0.2 + 0.09

Combine like terms: 0.2 + 0.09 = 0.29
0.09 + 0.6x + x2 = 0.29

Factor a perfect square on the left side:
(x + 0.3)(x + 0.3) = 0.29

Calculate the square root of the right side: 0.538516481

Break this problem into two subproblems by setting 
(x + 0.3) equal to 0.538516481 and -0.538516481.

Subproblem 1

x + 0.3 = 0.538516481 Simplifying x + 0.3 = 0.538516481 Reorder the terms: 0.3 + x = 0.538516481 Solving 0.3 + x = 0.538516481 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3' to each side of the equation. 0.3 + -0.3 + x = 0.538516481 + -0.3 Combine like terms: 0.3 + -0.3 = 0.0 0.0 + x = 0.538516481 + -0.3 x = 0.538516481 + -0.3 Combine like terms: 0.538516481 + -0.3 = 0.238516481 x = 0.238516481 Simplifying x = 0.238516481

Subproblem 2

x + 0.3 = -0.538516481 Simplifying x + 0.3 = -0.538516481 Reorder the terms: 0.3 + x = -0.538516481 Solving 0.3 + x = -0.538516481 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3' to each side of the equation. 0.3 + -0.3 + x = -0.538516481 + -0.3 Combine like terms: 0.3 + -0.3 = 0.0 0.0 + x = -0.538516481 + -0.3 x = -0.538516481 + -0.3 Combine like terms: -0.538516481 + -0.3 = -0.838516481 x = -0.838516481 Simplifying x = -0.838516481

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.238516481, -0.838516481}

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