5x^2-45x+84=0

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Solution for 5x^2-45x+84=0 equation:


Simplifying
5x2 + -45x + 84 = 0

Reorder the terms:
84 + -45x + 5x2 = 0

Solving
84 + -45x + 5x2 = 0

Solving for variable 'x'.

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

Divide each side by '5'.
16.8 + -9x + x2 = 0

Move the constant term to the right:

Add '-16.8' to each side of the equation.
16.8 + -9x + -16.8 + x2 = 0 + -16.8

Reorder the terms:
16.8 + -16.8 + -9x + x2 = 0 + -16.8

Combine like terms: 16.8 + -16.8 = 0.0
0.0 + -9x + x2 = 0 + -16.8
-9x + x2 = 0 + -16.8

Combine like terms: 0 + -16.8 = -16.8
-9x + x2 = -16.8

The x term is -9x.  Take half its coefficient (-4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
-9x + 20.25 + x2 = -16.8 + 20.25

Reorder the terms:
20.25 + -9x + x2 = -16.8 + 20.25

Combine like terms: -16.8 + 20.25 = 3.45
20.25 + -9x + x2 = 3.45

Factor a perfect square on the left side:
(x + -4.5)(x + -4.5) = 3.45

Calculate the square root of the right side: 1.857417562

Break this problem into two subproblems by setting 
(x + -4.5) equal to 1.857417562 and -1.857417562.

Subproblem 1

x + -4.5 = 1.857417562 Simplifying x + -4.5 = 1.857417562 Reorder the terms: -4.5 + x = 1.857417562 Solving -4.5 + x = 1.857417562 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + x = 1.857417562 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + x = 1.857417562 + 4.5 x = 1.857417562 + 4.5 Combine like terms: 1.857417562 + 4.5 = 6.357417562 x = 6.357417562 Simplifying x = 6.357417562

Subproblem 2

x + -4.5 = -1.857417562 Simplifying x + -4.5 = -1.857417562 Reorder the terms: -4.5 + x = -1.857417562 Solving -4.5 + x = -1.857417562 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + x = -1.857417562 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + x = -1.857417562 + 4.5 x = -1.857417562 + 4.5 Combine like terms: -1.857417562 + 4.5 = 2.642582438 x = 2.642582438 Simplifying x = 2.642582438

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6.357417562, 2.642582438}

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