5x^2-9=-20.5x

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Solution for 5x^2-9=-20.5x equation:


Simplifying
5x2 + -9 = -20.5x

Reorder the terms:
-9 + 5x2 = -20.5x

Solving
-9 + 5x2 = -20.5x

Solving for variable 'x'.

Reorder the terms:
-9 + 20.5x + 5x2 = -20.5x + 20.5x

Combine like terms: -20.5x + 20.5x = 0.0
-9 + 20.5x + 5x2 = 0.0

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

Divide each side by '5'.
-1.8 + 4.1x + x2 = 0.0

Move the constant term to the right:

Add '1.8' to each side of the equation.
-1.8 + 4.1x + 1.8 + x2 = 0.0 + 1.8

Reorder the terms:
-1.8 + 1.8 + 4.1x + x2 = 0.0 + 1.8

Combine like terms: -1.8 + 1.8 = 0.0
0.0 + 4.1x + x2 = 0.0 + 1.8
4.1x + x2 = 0.0 + 1.8

Combine like terms: 0.0 + 1.8 = 1.8
4.1x + x2 = 1.8

The x term is 4.1x.  Take half its coefficient (2.05).
Square it (4.2025) and add it to both sides.

Add '4.2025' to each side of the equation.
4.1x + 4.2025 + x2 = 1.8 + 4.2025

Reorder the terms:
4.2025 + 4.1x + x2 = 1.8 + 4.2025

Combine like terms: 1.8 + 4.2025 = 6.0025
4.2025 + 4.1x + x2 = 6.0025

Factor a perfect square on the left side:
(x + 2.05)(x + 2.05) = 6.0025

Calculate the square root of the right side: 2.45

Break this problem into two subproblems by setting 
(x + 2.05) equal to 2.45 and -2.45.

Subproblem 1

x + 2.05 = 2.45 Simplifying x + 2.05 = 2.45 Reorder the terms: 2.05 + x = 2.45 Solving 2.05 + x = 2.45 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.05' to each side of the equation. 2.05 + -2.05 + x = 2.45 + -2.05 Combine like terms: 2.05 + -2.05 = 0.00 0.00 + x = 2.45 + -2.05 x = 2.45 + -2.05 Combine like terms: 2.45 + -2.05 = 0.4 x = 0.4 Simplifying x = 0.4

Subproblem 2

x + 2.05 = -2.45 Simplifying x + 2.05 = -2.45 Reorder the terms: 2.05 + x = -2.45 Solving 2.05 + x = -2.45 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.05' to each side of the equation. 2.05 + -2.05 + x = -2.45 + -2.05 Combine like terms: 2.05 + -2.05 = 0.00 0.00 + x = -2.45 + -2.05 x = -2.45 + -2.05 Combine like terms: -2.45 + -2.05 = -4.5 x = -4.5 Simplifying x = -4.5

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.4, -4.5}

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