0.2x+1=-0.4x^2+12

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Solution for 0.2x+1=-0.4x^2+12 equation:


Simplifying
0.2x + 1 = -0.4x2 + 12

Reorder the terms:
1 + 0.2x = -0.4x2 + 12

Reorder the terms:
1 + 0.2x = 12 + -0.4x2

Solving
1 + 0.2x = 12 + -0.4x2

Solving for variable 'x'.

Reorder the terms:
1 + -12 + 0.2x + 0.4x2 = 12 + -0.4x2 + -12 + 0.4x2

Combine like terms: 1 + -12 = -11
-11 + 0.2x + 0.4x2 = 12 + -0.4x2 + -12 + 0.4x2

Reorder the terms:
-11 + 0.2x + 0.4x2 = 12 + -12 + -0.4x2 + 0.4x2

Combine like terms: 12 + -12 = 0
-11 + 0.2x + 0.4x2 = 0 + -0.4x2 + 0.4x2
-11 + 0.2x + 0.4x2 = -0.4x2 + 0.4x2

Combine like terms: -0.4x2 + 0.4x2 = 0.0
-11 + 0.2x + 0.4x2 = 0.0

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

Divide each side by '0.4'.
-27.5 + 0.5x + x2 = 0

Move the constant term to the right:

Add '27.5' to each side of the equation.
-27.5 + 0.5x + 27.5 + x2 = 0 + 27.5

Reorder the terms:
-27.5 + 27.5 + 0.5x + x2 = 0 + 27.5

Combine like terms: -27.5 + 27.5 = 0.0
0.0 + 0.5x + x2 = 0 + 27.5
0.5x + x2 = 0 + 27.5

Combine like terms: 0 + 27.5 = 27.5
0.5x + x2 = 27.5

The x term is 0.5x.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5x + 0.0625 + x2 = 27.5 + 0.0625

Reorder the terms:
0.0625 + 0.5x + x2 = 27.5 + 0.0625

Combine like terms: 27.5 + 0.0625 = 27.5625
0.0625 + 0.5x + x2 = 27.5625

Factor a perfect square on the left side:
(x + 0.25)(x + 0.25) = 27.5625

Calculate the square root of the right side: 5.25

Break this problem into two subproblems by setting 
(x + 0.25) equal to 5.25 and -5.25.

Subproblem 1

x + 0.25 = 5.25 Simplifying x + 0.25 = 5.25 Reorder the terms: 0.25 + x = 5.25 Solving 0.25 + x = 5.25 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = 5.25 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 5.25 + -0.25 x = 5.25 + -0.25 Combine like terms: 5.25 + -0.25 = 5 x = 5 Simplifying x = 5

Subproblem 2

x + 0.25 = -5.25 Simplifying x + 0.25 = -5.25 Reorder the terms: 0.25 + x = -5.25 Solving 0.25 + x = -5.25 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = -5.25 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -5.25 + -0.25 x = -5.25 + -0.25 Combine like terms: -5.25 + -0.25 = -5.5 x = -5.5 Simplifying x = -5.5

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5, -5.5}

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