x^2+5x=45

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Solution for x^2+5x=45 equation:


Simplifying
x2 + 5x = 45

Reorder the terms:
5x + x2 = 45

Solving
5x + x2 = 45

Solving for variable 'x'.

Reorder the terms:
-45 + 5x + x2 = 45 + -45

Combine like terms: 45 + -45 = 0
-45 + 5x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '45' to each side of the equation.
-45 + 5x + 45 + x2 = 0 + 45

Reorder the terms:
-45 + 45 + 5x + x2 = 0 + 45

Combine like terms: -45 + 45 = 0
0 + 5x + x2 = 0 + 45
5x + x2 = 0 + 45

Combine like terms: 0 + 45 = 45
5x + x2 = 45

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 = 45 + 6.25

Reorder the terms:
6.25 + 5x + x2 = 45 + 6.25

Combine like terms: 45 + 6.25 = 51.25
6.25 + 5x + x2 = 51.25

Factor a perfect square on the left side:
(x + 2.5)(x + 2.5) = 51.25

Calculate the square root of the right side: 7.158910532

Break this problem into two subproblems by setting 
(x + 2.5) equal to 7.158910532 and -7.158910532.

Subproblem 1

x + 2.5 = 7.158910532 Simplifying x + 2.5 = 7.158910532 Reorder the terms: 2.5 + x = 7.158910532 Solving 2.5 + x = 7.158910532 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 = 7.158910532 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = 7.158910532 + -2.5 x = 7.158910532 + -2.5 Combine like terms: 7.158910532 + -2.5 = 4.658910532 x = 4.658910532 Simplifying x = 4.658910532

Subproblem 2

x + 2.5 = -7.158910532 Simplifying x + 2.5 = -7.158910532 Reorder the terms: 2.5 + x = -7.158910532 Solving 2.5 + x = -7.158910532 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 = -7.158910532 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = -7.158910532 + -2.5 x = -7.158910532 + -2.5 Combine like terms: -7.158910532 + -2.5 = -9.658910532 x = -9.658910532 Simplifying x = -9.658910532

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.658910532, -9.658910532}

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