5x-3=x^2-1

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


Simplifying
5x + -3 = x2 + -1

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

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

Solving
-3 + 5x = -1 + x2

Solving for variable 'x'.

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

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

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

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

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

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
2 + -5x + x2 = 0

Move the constant term to the right:

Add '-2' to each side of the equation.
2 + -5x + -2 + x2 = 0 + -2

Reorder the terms:
2 + -2 + -5x + x2 = 0 + -2

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

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

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 = -2 + 6.25

Reorder the terms:
6.25 + -5x + x2 = -2 + 6.25

Combine like terms: -2 + 6.25 = 4.25
6.25 + -5x + x2 = 4.25

Factor a perfect square on the left side:
(x + -2.5)(x + -2.5) = 4.25

Calculate the square root of the right side: 2.061552813

Break this problem into two subproblems by setting 
(x + -2.5) equal to 2.061552813 and -2.061552813.

Subproblem 1

x + -2.5 = 2.061552813 Simplifying x + -2.5 = 2.061552813 Reorder the terms: -2.5 + x = 2.061552813 Solving -2.5 + x = 2.061552813 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 = 2.061552813 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = 2.061552813 + 2.5 x = 2.061552813 + 2.5 Combine like terms: 2.061552813 + 2.5 = 4.561552813 x = 4.561552813 Simplifying x = 4.561552813

Subproblem 2

x + -2.5 = -2.061552813 Simplifying x + -2.5 = -2.061552813 Reorder the terms: -2.5 + x = -2.061552813 Solving -2.5 + x = -2.061552813 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 = -2.061552813 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = -2.061552813 + 2.5 x = -2.061552813 + 2.5 Combine like terms: -2.061552813 + 2.5 = 0.438447187 x = 0.438447187 Simplifying x = 0.438447187

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.561552813, 0.438447187}

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