x^2+45x=90

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Solution for x^2+45x=90 equation:


Simplifying
x2 + 45x = 90

Reorder the terms:
45x + x2 = 90

Solving
45x + x2 = 90

Solving for variable 'x'.

Reorder the terms:
-90 + 45x + x2 = 90 + -90

Combine like terms: 90 + -90 = 0
-90 + 45x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '90' to each side of the equation.
-90 + 45x + 90 + x2 = 0 + 90

Reorder the terms:
-90 + 90 + 45x + x2 = 0 + 90

Combine like terms: -90 + 90 = 0
0 + 45x + x2 = 0 + 90
45x + x2 = 0 + 90

Combine like terms: 0 + 90 = 90
45x + x2 = 90

The x term is 45x.  Take half its coefficient (22.5).
Square it (506.25) and add it to both sides.

Add '506.25' to each side of the equation.
45x + 506.25 + x2 = 90 + 506.25

Reorder the terms:
506.25 + 45x + x2 = 90 + 506.25

Combine like terms: 90 + 506.25 = 596.25
506.25 + 45x + x2 = 596.25

Factor a perfect square on the left side:
(x + 22.5)(x + 22.5) = 596.25

Calculate the square root of the right side: 24.418230894

Break this problem into two subproblems by setting 
(x + 22.5) equal to 24.418230894 and -24.418230894.

Subproblem 1

x + 22.5 = 24.418230894 Simplifying x + 22.5 = 24.418230894 Reorder the terms: 22.5 + x = 24.418230894 Solving 22.5 + x = 24.418230894 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-22.5' to each side of the equation. 22.5 + -22.5 + x = 24.418230894 + -22.5 Combine like terms: 22.5 + -22.5 = 0.0 0.0 + x = 24.418230894 + -22.5 x = 24.418230894 + -22.5 Combine like terms: 24.418230894 + -22.5 = 1.918230894 x = 1.918230894 Simplifying x = 1.918230894

Subproblem 2

x + 22.5 = -24.418230894 Simplifying x + 22.5 = -24.418230894 Reorder the terms: 22.5 + x = -24.418230894 Solving 22.5 + x = -24.418230894 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-22.5' to each side of the equation. 22.5 + -22.5 + x = -24.418230894 + -22.5 Combine like terms: 22.5 + -22.5 = 0.0 0.0 + x = -24.418230894 + -22.5 x = -24.418230894 + -22.5 Combine like terms: -24.418230894 + -22.5 = -46.918230894 x = -46.918230894 Simplifying x = -46.918230894

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.918230894, -46.918230894}

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