-.5x^2+4x+1=5

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Solution for -.5x^2+4x+1=5 equation:


Simplifying
-0.5x2 + 4x + 1 = 5

Reorder the terms:
1 + 4x + -0.5x2 = 5

Solving
1 + 4x + -0.5x2 = 5

Solving for variable 'x'.

Reorder the terms:
1 + -5 + 4x + -0.5x2 = 5 + -5

Combine like terms: 1 + -5 = -4
-4 + 4x + -0.5x2 = 5 + -5

Combine like terms: 5 + -5 = 0
-4 + 4x + -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'.
8 + -8x + x2 = 0

Move the constant term to the right:

Add '-8' to each side of the equation.
8 + -8x + -8 + x2 = 0 + -8

Reorder the terms:
8 + -8 + -8x + x2 = 0 + -8

Combine like terms: 8 + -8 = 0
0 + -8x + x2 = 0 + -8
-8x + x2 = 0 + -8

Combine like terms: 0 + -8 = -8
-8x + x2 = -8

The x term is -8x.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8x + 16 + x2 = -8 + 16

Reorder the terms:
16 + -8x + x2 = -8 + 16

Combine like terms: -8 + 16 = 8
16 + -8x + x2 = 8

Factor a perfect square on the left side:
(x + -4)(x + -4) = 8

Calculate the square root of the right side: 2.828427125

Break this problem into two subproblems by setting 
(x + -4) equal to 2.828427125 and -2.828427125.

Subproblem 1

x + -4 = 2.828427125 Simplifying x + -4 = 2.828427125 Reorder the terms: -4 + x = 2.828427125 Solving -4 + x = 2.828427125 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + x = 2.828427125 + 4 Combine like terms: -4 + 4 = 0 0 + x = 2.828427125 + 4 x = 2.828427125 + 4 Combine like terms: 2.828427125 + 4 = 6.828427125 x = 6.828427125 Simplifying x = 6.828427125

Subproblem 2

x + -4 = -2.828427125 Simplifying x + -4 = -2.828427125 Reorder the terms: -4 + x = -2.828427125 Solving -4 + x = -2.828427125 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + x = -2.828427125 + 4 Combine like terms: -4 + 4 = 0 0 + x = -2.828427125 + 4 x = -2.828427125 + 4 Combine like terms: -2.828427125 + 4 = 1.171572875 x = 1.171572875 Simplifying x = 1.171572875

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6.828427125, 1.171572875}

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