45=3x+x^2

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Solution for 45=3x+x^2 equation:


Simplifying
45 = 3x + x2

Solving
45 = 3x + x2

Solving for variable 'x'.

Reorder the terms:
45 + -3x + -1x2 = 3x + -3x + x2 + -1x2

Combine like terms: 3x + -3x = 0
45 + -3x + -1x2 = 0 + x2 + -1x2
45 + -3x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
45 + -3x + -1x2 = 0

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

Divide each side by '-1'.
-45 + 3x + x2 = 0

Move the constant term to the right:

Add '45' to each side of the equation.
-45 + 3x + 45 + x2 = 0 + 45

Reorder the terms:
-45 + 45 + 3x + x2 = 0 + 45

Combine like terms: -45 + 45 = 0
0 + 3x + x2 = 0 + 45
3x + x2 = 0 + 45

Combine like terms: 0 + 45 = 45
3x + x2 = 45

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 = 45 + 2.25

Reorder the terms:
2.25 + 3x + x2 = 45 + 2.25

Combine like terms: 45 + 2.25 = 47.25
2.25 + 3x + x2 = 47.25

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 47.25

Calculate the square root of the right side: 6.873863542

Break this problem into two subproblems by setting 
(x + 1.5) equal to 6.873863542 and -6.873863542.

Subproblem 1

x + 1.5 = 6.873863542 Simplifying x + 1.5 = 6.873863542 Reorder the terms: 1.5 + x = 6.873863542 Solving 1.5 + x = 6.873863542 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 = 6.873863542 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 6.873863542 + -1.5 x = 6.873863542 + -1.5 Combine like terms: 6.873863542 + -1.5 = 5.373863542 x = 5.373863542 Simplifying x = 5.373863542

Subproblem 2

x + 1.5 = -6.873863542 Simplifying x + 1.5 = -6.873863542 Reorder the terms: 1.5 + x = -6.873863542 Solving 1.5 + x = -6.873863542 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 = -6.873863542 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -6.873863542 + -1.5 x = -6.873863542 + -1.5 Combine like terms: -6.873863542 + -1.5 = -8.373863542 x = -8.373863542 Simplifying x = -8.373863542

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.373863542, -8.373863542}

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