4x-13=x^2-23

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Solution for 4x-13=x^2-23 equation:


Simplifying
4x + -13 = x2 + -23

Reorder the terms:
-13 + 4x = x2 + -23

Reorder the terms:
-13 + 4x = -23 + x2

Solving
-13 + 4x = -23 + x2

Solving for variable 'x'.

Reorder the terms:
-13 + 23 + 4x + -1x2 = -23 + x2 + 23 + -1x2

Combine like terms: -13 + 23 = 10
10 + 4x + -1x2 = -23 + x2 + 23 + -1x2

Reorder the terms:
10 + 4x + -1x2 = -23 + 23 + x2 + -1x2

Combine like terms: -23 + 23 = 0
10 + 4x + -1x2 = 0 + x2 + -1x2
10 + 4x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
10 + 4x + -1x2 = 0

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

Divide each side by '-1'.
-10 + -4x + x2 = 0

Move the constant term to the right:

Add '10' to each side of the equation.
-10 + -4x + 10 + x2 = 0 + 10

Reorder the terms:
-10 + 10 + -4x + x2 = 0 + 10

Combine like terms: -10 + 10 = 0
0 + -4x + x2 = 0 + 10
-4x + x2 = 0 + 10

Combine like terms: 0 + 10 = 10
-4x + x2 = 10

The x term is -4x.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
-4x + 4 + x2 = 10 + 4

Reorder the terms:
4 + -4x + x2 = 10 + 4

Combine like terms: 10 + 4 = 14
4 + -4x + x2 = 14

Factor a perfect square on the left side:
(x + -2)(x + -2) = 14

Calculate the square root of the right side: 3.741657387

Break this problem into two subproblems by setting 
(x + -2) equal to 3.741657387 and -3.741657387.

Subproblem 1

x + -2 = 3.741657387 Simplifying x + -2 = 3.741657387 Reorder the terms: -2 + x = 3.741657387 Solving -2 + x = 3.741657387 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + x = 3.741657387 + 2 Combine like terms: -2 + 2 = 0 0 + x = 3.741657387 + 2 x = 3.741657387 + 2 Combine like terms: 3.741657387 + 2 = 5.741657387 x = 5.741657387 Simplifying x = 5.741657387

Subproblem 2

x + -2 = -3.741657387 Simplifying x + -2 = -3.741657387 Reorder the terms: -2 + x = -3.741657387 Solving -2 + x = -3.741657387 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + x = -3.741657387 + 2 Combine like terms: -2 + 2 = 0 0 + x = -3.741657387 + 2 x = -3.741657387 + 2 Combine like terms: -3.741657387 + 2 = -1.741657387 x = -1.741657387 Simplifying x = -1.741657387

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.741657387, -1.741657387}

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