.5x^2+.5x+1=37

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Solution for .5x^2+.5x+1=37 equation:


Simplifying
0.5x2 + 0.5x + 1 = 37

Reorder the terms:
1 + 0.5x + 0.5x2 = 37

Solving
1 + 0.5x + 0.5x2 = 37

Solving for variable 'x'.

Reorder the terms:
1 + -37 + 0.5x + 0.5x2 = 37 + -37

Combine like terms: 1 + -37 = -36
-36 + 0.5x + 0.5x2 = 37 + -37

Combine like terms: 37 + -37 = 0
-36 + 0.5x + 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'.
-72 + x + x2 = 0

Move the constant term to the right:

Add '72' to each side of the equation.
-72 + x + 72 + x2 = 0 + 72

Reorder the terms:
-72 + 72 + x + x2 = 0 + 72

Combine like terms: -72 + 72 = 0
0 + x + x2 = 0 + 72
x + x2 = 0 + 72

Combine like terms: 0 + 72 = 72
x + x2 = 72

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.
 + 0.25 + x2 = 72 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + x2 = 72 + 0.25

Combine like terms: 72 + 0.25 = 72.25
1.25 + x2 = 72.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 72.25

Calculate the square root of the right side: 8.5

Break this problem into two subproblems by setting 
(x + 0.5) equal to 8.5 and -8.5.

Subproblem 1

x + 0.5 = 8.5 Simplifying x + 0.5 = 8.5 Reorder the terms: 0.5 + x = 8.5 Solving 0.5 + x = 8.5 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 = 8.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 8.5 + -0.5 x = 8.5 + -0.5 Combine like terms: 8.5 + -0.5 = 8 x = 8 Simplifying x = 8

Subproblem 2

x + 0.5 = -8.5 Simplifying x + 0.5 = -8.5 Reorder the terms: 0.5 + x = -8.5 Solving 0.5 + x = -8.5 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 = -8.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -8.5 + -0.5 x = -8.5 + -0.5 Combine like terms: -8.5 + -0.5 = -9 x = -9 Simplifying x = -9

Solution

The solution to the problem is based on the solutions from the subproblems. x = {8, -9}

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