45x+78=x^2

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Solution for 45x+78=x^2 equation:


Simplifying
45x + 78 = x2

Reorder the terms:
78 + 45x = x2

Solving
78 + 45x = x2

Solving for variable 'x'.

Combine like terms: x2 + -1x2 = 0
78 + 45x + -1x2 = 0

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

Divide each side by '-1'.
-78 + -45x + x2 = 0

Move the constant term to the right:

Add '78' to each side of the equation.
-78 + -45x + 78 + x2 = 0 + 78

Reorder the terms:
-78 + 78 + -45x + x2 = 0 + 78

Combine like terms: -78 + 78 = 0
0 + -45x + x2 = 0 + 78
-45x + x2 = 0 + 78

Combine like terms: 0 + 78 = 78
-45x + x2 = 78

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 = 78 + 506.25

Reorder the terms:
506.25 + -45x + x2 = 78 + 506.25

Combine like terms: 78 + 506.25 = 584.25
506.25 + -45x + x2 = 584.25

Factor a perfect square on the left side:
(x + -22.5)(x + -22.5) = 584.25

Calculate the square root of the right side: 24.171263931

Break this problem into two subproblems by setting 
(x + -22.5) equal to 24.171263931 and -24.171263931.

Subproblem 1

x + -22.5 = 24.171263931 Simplifying x + -22.5 = 24.171263931 Reorder the terms: -22.5 + x = 24.171263931 Solving -22.5 + x = 24.171263931 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 = 24.171263931 + 22.5 Combine like terms: -22.5 + 22.5 = 0.0 0.0 + x = 24.171263931 + 22.5 x = 24.171263931 + 22.5 Combine like terms: 24.171263931 + 22.5 = 46.671263931 x = 46.671263931 Simplifying x = 46.671263931

Subproblem 2

x + -22.5 = -24.171263931 Simplifying x + -22.5 = -24.171263931 Reorder the terms: -22.5 + x = -24.171263931 Solving -22.5 + x = -24.171263931 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 = -24.171263931 + 22.5 Combine like terms: -22.5 + 22.5 = 0.0 0.0 + x = -24.171263931 + 22.5 x = -24.171263931 + 22.5 Combine like terms: -24.171263931 + 22.5 = -1.671263931 x = -1.671263931 Simplifying x = -1.671263931

Solution

The solution to the problem is based on the solutions from the subproblems. x = {46.671263931, -1.671263931}

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