4x=x^2+13x-5

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Solution for 4x=x^2+13x-5 equation:


Simplifying
4x = x2 + 13x + -5

Reorder the terms:
4x = -5 + 13x + x2

Solving
4x = -5 + 13x + x2

Solving for variable 'x'.

Reorder the terms:
5 + 4x + -13x + -1x2 = -5 + 13x + x2 + 5 + -13x + -1x2

Combine like terms: 4x + -13x = -9x
5 + -9x + -1x2 = -5 + 13x + x2 + 5 + -13x + -1x2

Reorder the terms:
5 + -9x + -1x2 = -5 + 5 + 13x + -13x + x2 + -1x2

Combine like terms: -5 + 5 = 0
5 + -9x + -1x2 = 0 + 13x + -13x + x2 + -1x2
5 + -9x + -1x2 = 13x + -13x + x2 + -1x2

Combine like terms: 13x + -13x = 0
5 + -9x + -1x2 = 0 + x2 + -1x2
5 + -9x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
5 + -9x + -1x2 = 0

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

Divide each side by '-1'.
-5 + 9x + x2 = 0

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + 9x + 5 + x2 = 0 + 5

Reorder the terms:
-5 + 5 + 9x + x2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + 9x + x2 = 0 + 5
9x + x2 = 0 + 5

Combine like terms: 0 + 5 = 5
9x + x2 = 5

The x term is 9x.  Take half its coefficient (4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
9x + 20.25 + x2 = 5 + 20.25

Reorder the terms:
20.25 + 9x + x2 = 5 + 20.25

Combine like terms: 5 + 20.25 = 25.25
20.25 + 9x + x2 = 25.25

Factor a perfect square on the left side:
(x + 4.5)(x + 4.5) = 25.25

Calculate the square root of the right side: 5.024937811

Break this problem into two subproblems by setting 
(x + 4.5) equal to 5.024937811 and -5.024937811.

Subproblem 1

x + 4.5 = 5.024937811 Simplifying x + 4.5 = 5.024937811 Reorder the terms: 4.5 + x = 5.024937811 Solving 4.5 + x = 5.024937811 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + x = 5.024937811 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + x = 5.024937811 + -4.5 x = 5.024937811 + -4.5 Combine like terms: 5.024937811 + -4.5 = 0.524937811 x = 0.524937811 Simplifying x = 0.524937811

Subproblem 2

x + 4.5 = -5.024937811 Simplifying x + 4.5 = -5.024937811 Reorder the terms: 4.5 + x = -5.024937811 Solving 4.5 + x = -5.024937811 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + x = -5.024937811 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + x = -5.024937811 + -4.5 x = -5.024937811 + -4.5 Combine like terms: -5.024937811 + -4.5 = -9.524937811 x = -9.524937811 Simplifying x = -9.524937811

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.524937811, -9.524937811}

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