x^2+5x=3x+1

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


Simplifying
x2 + 5x = 3x + 1

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

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

Solving
5x + x2 = 1 + 3x

Solving for variable 'x'.

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

Combine like terms: 5x + -3x = 2x
-1 + 2x + x2 = 1 + 3x + -1 + -3x

Reorder the terms:
-1 + 2x + x2 = 1 + -1 + 3x + -3x

Combine like terms: 1 + -1 = 0
-1 + 2x + x2 = 0 + 3x + -3x
-1 + 2x + x2 = 3x + -3x

Combine like terms: 3x + -3x = 0
-1 + 2x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 2x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + 2x + x2 = 0 + 1

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

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

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 = 1 + 1

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

Combine like terms: 1 + 1 = 2
1 + 2x + x2 = 2

Factor a perfect square on the left side:
(x + 1)(x + 1) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(x + 1) equal to 1.414213562 and -1.414213562.

Subproblem 1

x + 1 = 1.414213562 Simplifying x + 1 = 1.414213562 Reorder the terms: 1 + x = 1.414213562 Solving 1 + x = 1.414213562 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 = 1.414213562 + -1 Combine like terms: 1 + -1 = 0 0 + x = 1.414213562 + -1 x = 1.414213562 + -1 Combine like terms: 1.414213562 + -1 = 0.414213562 x = 0.414213562 Simplifying x = 0.414213562

Subproblem 2

x + 1 = -1.414213562 Simplifying x + 1 = -1.414213562 Reorder the terms: 1 + x = -1.414213562 Solving 1 + x = -1.414213562 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 = -1.414213562 + -1 Combine like terms: 1 + -1 = 0 0 + x = -1.414213562 + -1 x = -1.414213562 + -1 Combine like terms: -1.414213562 + -1 = -2.414213562 x = -2.414213562 Simplifying x = -2.414213562

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.414213562, -2.414213562}

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