x^2=3x+13

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Solution for x^2=3x+13 equation:


Simplifying
x2 = 3x + 13

Reorder the terms:
x2 = 13 + 3x

Solving
x2 = 13 + 3x

Solving for variable 'x'.

Reorder the terms:
-13 + -3x + x2 = 13 + 3x + -13 + -3x

Reorder the terms:
-13 + -3x + x2 = 13 + -13 + 3x + -3x

Combine like terms: 13 + -13 = 0
-13 + -3x + x2 = 0 + 3x + -3x
-13 + -3x + x2 = 3x + -3x

Combine like terms: 3x + -3x = 0
-13 + -3x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '13' to each side of the equation.
-13 + -3x + 13 + x2 = 0 + 13

Reorder the terms:
-13 + 13 + -3x + x2 = 0 + 13

Combine like terms: -13 + 13 = 0
0 + -3x + x2 = 0 + 13
-3x + x2 = 0 + 13

Combine like terms: 0 + 13 = 13
-3x + x2 = 13

The x term is -3x.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3x + 2.25 + x2 = 13 + 2.25

Reorder the terms:
2.25 + -3x + x2 = 13 + 2.25

Combine like terms: 13 + 2.25 = 15.25
2.25 + -3x + x2 = 15.25

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 15.25

Calculate the square root of the right side: 3.905124838

Break this problem into two subproblems by setting 
(x + -1.5) equal to 3.905124838 and -3.905124838.

Subproblem 1

x + -1.5 = 3.905124838 Simplifying x + -1.5 = 3.905124838 Reorder the terms: -1.5 + x = 3.905124838 Solving -1.5 + x = 3.905124838 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = 3.905124838 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 3.905124838 + 1.5 x = 3.905124838 + 1.5 Combine like terms: 3.905124838 + 1.5 = 5.405124838 x = 5.405124838 Simplifying x = 5.405124838

Subproblem 2

x + -1.5 = -3.905124838 Simplifying x + -1.5 = -3.905124838 Reorder the terms: -1.5 + x = -3.905124838 Solving -1.5 + x = -3.905124838 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = -3.905124838 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -3.905124838 + 1.5 x = -3.905124838 + 1.5 Combine like terms: -3.905124838 + 1.5 = -2.405124838 x = -2.405124838 Simplifying x = -2.405124838

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.405124838, -2.405124838}

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