x^2+5x=31

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Solution for x^2+5x=31 equation:


Simplifying
x2 + 5x = 31

Reorder the terms:
5x + x2 = 31

Solving
5x + x2 = 31

Solving for variable 'x'.

Reorder the terms:
-31 + 5x + x2 = 31 + -31

Combine like terms: 31 + -31 = 0
-31 + 5x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '31' to each side of the equation.
-31 + 5x + 31 + x2 = 0 + 31

Reorder the terms:
-31 + 31 + 5x + x2 = 0 + 31

Combine like terms: -31 + 31 = 0
0 + 5x + x2 = 0 + 31
5x + x2 = 0 + 31

Combine like terms: 0 + 31 = 31
5x + x2 = 31

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 = 31 + 6.25

Reorder the terms:
6.25 + 5x + x2 = 31 + 6.25

Combine like terms: 31 + 6.25 = 37.25
6.25 + 5x + x2 = 37.25

Factor a perfect square on the left side:
(x + 2.5)(x + 2.5) = 37.25

Calculate the square root of the right side: 6.103277808

Break this problem into two subproblems by setting 
(x + 2.5) equal to 6.103277808 and -6.103277808.

Subproblem 1

x + 2.5 = 6.103277808 Simplifying x + 2.5 = 6.103277808 Reorder the terms: 2.5 + x = 6.103277808 Solving 2.5 + x = 6.103277808 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 = 6.103277808 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = 6.103277808 + -2.5 x = 6.103277808 + -2.5 Combine like terms: 6.103277808 + -2.5 = 3.603277808 x = 3.603277808 Simplifying x = 3.603277808

Subproblem 2

x + 2.5 = -6.103277808 Simplifying x + 2.5 = -6.103277808 Reorder the terms: 2.5 + x = -6.103277808 Solving 2.5 + x = -6.103277808 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 = -6.103277808 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = -6.103277808 + -2.5 x = -6.103277808 + -2.5 Combine like terms: -6.103277808 + -2.5 = -8.603277808 x = -8.603277808 Simplifying x = -8.603277808

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.603277808, -8.603277808}

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