6x^2+1=5x-0.25x^2

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Solution for 6x^2+1=5x-0.25x^2 equation:


Simplifying
6x2 + 1 = 5x + -0.25x2

Reorder the terms:
1 + 6x2 = 5x + -0.25x2

Solving
1 + 6x2 = 5x + -0.25x2

Solving for variable 'x'.

Reorder the terms:
1 + -5x + 6x2 + 0.25x2 = 5x + -0.25x2 + -5x + 0.25x2

Combine like terms: 6x2 + 0.25x2 = 6.25x2
1 + -5x + 6.25x2 = 5x + -0.25x2 + -5x + 0.25x2

Reorder the terms:
1 + -5x + 6.25x2 = 5x + -5x + -0.25x2 + 0.25x2

Combine like terms: 5x + -5x = 0
1 + -5x + 6.25x2 = 0 + -0.25x2 + 0.25x2
1 + -5x + 6.25x2 = -0.25x2 + 0.25x2

Combine like terms: -0.25x2 + 0.25x2 = 0.00
1 + -5x + 6.25x2 = 0.00

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

Divide each side by '6.25'.
0.16 + -0.8x + x2 = 0

Move the constant term to the right:

Add '-0.16' to each side of the equation.
0.16 + -0.8x + -0.16 + x2 = 0 + -0.16

Reorder the terms:
0.16 + -0.16 + -0.8x + x2 = 0 + -0.16

Combine like terms: 0.16 + -0.16 = 0.00
0.00 + -0.8x + x2 = 0 + -0.16
-0.8x + x2 = 0 + -0.16

Combine like terms: 0 + -0.16 = -0.16
-0.8x + x2 = -0.16

The x term is -0.8x.  Take half its coefficient (-0.4).
Square it (0.16) and add it to both sides.

Add '0.16' to each side of the equation.
-0.8x + 0.16 + x2 = -0.16 + 0.16

Reorder the terms:
0.16 + -0.8x + x2 = -0.16 + 0.16

Combine like terms: -0.16 + 0.16 = 0.00
0.16 + -0.8x + x2 = 0.00

Factor a perfect square on the left side:
(x + -0.4)(x + -0.4) = 0.00

Calculate the square root of the right side: 0.00

Break this problem into two subproblems by setting 
(x + -0.4) equal to 0.00 and 0.00.

Subproblem 1

x + -0.4 = 0.00 Simplifying x + -0.4 = 0.00 Reorder the terms: -0.4 + x = 0.00 Solving -0.4 + x = 0.00 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4' to each side of the equation. -0.4 + 0.4 + x = 0.00 + 0.4 Combine like terms: -0.4 + 0.4 = 0.0 0.0 + x = 0.00 + 0.4 x = 0.00 + 0.4 Combine like terms: 0.00 + 0.4 = 0.4 x = 0.4 Simplifying x = 0.4

Subproblem 2

x + -0.4 = 0.00 Simplifying x + -0.4 = 0.00 Reorder the terms: -0.4 + x = 0.00 Solving -0.4 + x = 0.00 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4' to each side of the equation. -0.4 + 0.4 + x = 0.00 + 0.4 Combine like terms: -0.4 + 0.4 = 0.0 0.0 + x = 0.00 + 0.4 x = 0.00 + 0.4 Combine like terms: 0.00 + 0.4 = 0.4 x = 0.4 Simplifying x = 0.4

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.4, 0.4}

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