0.5x^2+x=40

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Solution for 0.5x^2+x=40 equation:


Simplifying
0.5x2 + x = 40

Reorder the terms:
x + 0.5x2 = 40

Solving
x + 0.5x2 = 40

Solving for variable 'x'.

Reorder the terms:
-40 + x + 0.5x2 = 40 + -40

Combine like terms: 40 + -40 = 0
-40 + x + 0.5x2 = 0

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

Divide each side by '0.5'.
-80 + 2x + x2 = 0

Move the constant term to the right:

Add '80' to each side of the equation.
-80 + 2x + 80 + x2 = 0 + 80

Reorder the terms:
-80 + 80 + 2x + x2 = 0 + 80

Combine like terms: -80 + 80 = 0
0 + 2x + x2 = 0 + 80
2x + x2 = 0 + 80

Combine like terms: 0 + 80 = 80
2x + x2 = 80

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
2x + 0.25 + x2 = 80 + 0.25

Reorder the terms:
0.25 + 2x + x2 = 80 + 0.25

Combine like terms: 80 + 0.25 = 80.25
0.25 + 2x + x2 = 80.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 80.25

Calculate the square root of the right side: 8.958236434

Break this problem into two subproblems by setting 
(x + 0.5) equal to 8.958236434 and -8.958236434.

Subproblem 1

x + 0.5 = 8.958236434 Simplifying x + 0.5 = 8.958236434 Reorder the terms: 0.5 + x = 8.958236434 Solving 0.5 + x = 8.958236434 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 8.958236434 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 8.958236434 + -0.5 x = 8.958236434 + -0.5 Combine like terms: 8.958236434 + -0.5 = 8.458236434 x = 8.458236434 Simplifying x = 8.458236434

Subproblem 2

x + 0.5 = -8.958236434 Simplifying x + 0.5 = -8.958236434 Reorder the terms: 0.5 + x = -8.958236434 Solving 0.5 + x = -8.958236434 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -8.958236434 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -8.958236434 + -0.5 x = -8.958236434 + -0.5 Combine like terms: -8.958236434 + -0.5 = -9.458236434 x = -9.458236434 Simplifying x = -9.458236434

Solution

The solution to the problem is based on the solutions from the subproblems. x = {8.458236434, -9.458236434}

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