x^2=5x-3

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Solution for x^2=5x-3 equation:


Simplifying
x2 = 5x + -3

Reorder the terms:
x2 = -3 + 5x

Solving
x2 = -3 + 5x

Solving for variable 'x'.

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

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

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

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

Begin completing the square.

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + -5x + -3 + x2 = 0 + -3

Reorder the terms:
3 + -3 + -5x + x2 = 0 + -3

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

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

The x term is -5x.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5x + 6.25 + x2 = -3 + 6.25

Reorder the terms:
6.25 + -5x + x2 = -3 + 6.25

Combine like terms: -3 + 6.25 = 3.25
6.25 + -5x + x2 = 3.25

Factor a perfect square on the left side:
(x + -2.5)(x + -2.5) = 3.25

Calculate the square root of the right side: 1.802775638

Break this problem into two subproblems by setting 
(x + -2.5) equal to 1.802775638 and -1.802775638.

Subproblem 1

x + -2.5 = 1.802775638 Simplifying x + -2.5 = 1.802775638 Reorder the terms: -2.5 + x = 1.802775638 Solving -2.5 + x = 1.802775638 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + x = 1.802775638 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = 1.802775638 + 2.5 x = 1.802775638 + 2.5 Combine like terms: 1.802775638 + 2.5 = 4.302775638 x = 4.302775638 Simplifying x = 4.302775638

Subproblem 2

x + -2.5 = -1.802775638 Simplifying x + -2.5 = -1.802775638 Reorder the terms: -2.5 + x = -1.802775638 Solving -2.5 + x = -1.802775638 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + x = -1.802775638 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = -1.802775638 + 2.5 x = -1.802775638 + 2.5 Combine like terms: -1.802775638 + 2.5 = 0.697224362 x = 0.697224362 Simplifying x = 0.697224362

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.302775638, 0.697224362}

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