.5x^2+x=10

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Solution for .5x^2+x=10 equation:


Simplifying
0.5x2 + x = 10

Reorder the terms:
x + 0.5x2 = 10

Solving
x + 0.5x2 = 10

Solving for variable 'x'.

Reorder the terms:
-10 + x + 0.5x2 = 10 + -10

Combine like terms: 10 + -10 = 0
-10 + 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'.
-20 + 2x + x2 = 0

Move the constant term to the right:

Add '20' to each side of the equation.
-20 + 2x + 20 + x2 = 0 + 20

Reorder the terms:
-20 + 20 + 2x + x2 = 0 + 20

Combine like terms: -20 + 20 = 0
0 + 2x + x2 = 0 + 20
2x + x2 = 0 + 20

Combine like terms: 0 + 20 = 20
2x + x2 = 20

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 = 20 + 0.25

Reorder the terms:
0.25 + 2x + x2 = 20 + 0.25

Combine like terms: 20 + 0.25 = 20.25
0.25 + 2x + x2 = 20.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 20.25

Calculate the square root of the right side: 4.5

Break this problem into two subproblems by setting 
(x + 0.5) equal to 4.5 and -4.5.

Subproblem 1

x + 0.5 = 4.5 Simplifying x + 0.5 = 4.5 Reorder the terms: 0.5 + x = 4.5 Solving 0.5 + x = 4.5 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 = 4.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 4.5 + -0.5 x = 4.5 + -0.5 Combine like terms: 4.5 + -0.5 = 4 x = 4 Simplifying x = 4

Subproblem 2

x + 0.5 = -4.5 Simplifying x + 0.5 = -4.5 Reorder the terms: 0.5 + x = -4.5 Solving 0.5 + x = -4.5 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 = -4.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -4.5 + -0.5 x = -4.5 + -0.5 Combine like terms: -4.5 + -0.5 = -5 x = -5 Simplifying x = -5

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4, -5}

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