0.25x^2+6.25+2.5x=25-x^2

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Solution for 0.25x^2+6.25+2.5x=25-x^2 equation:


Simplifying
0.25x2 + 6.25 + 2.5x = 25 + -1x2

Reorder the terms:
6.25 + 2.5x + 0.25x2 = 25 + -1x2

Solving
6.25 + 2.5x + 0.25x2 = 25 + -1x2

Solving for variable 'x'.

Reorder the terms:
6.25 + -25 + 2.5x + 0.25x2 + x2 = 25 + -1x2 + -25 + x2

Combine like terms: 6.25 + -25 = -18.75
-18.75 + 2.5x + 0.25x2 + x2 = 25 + -1x2 + -25 + x2

Combine like terms: 0.25x2 + x2 = 1.25x2
-18.75 + 2.5x + 1.25x2 = 25 + -1x2 + -25 + x2

Reorder the terms:
-18.75 + 2.5x + 1.25x2 = 25 + -25 + -1x2 + x2

Combine like terms: 25 + -25 = 0
-18.75 + 2.5x + 1.25x2 = 0 + -1x2 + x2
-18.75 + 2.5x + 1.25x2 = -1x2 + x2

Combine like terms: -1x2 + x2 = 0
-18.75 + 2.5x + 1.25x2 = 0

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

Divide each side by '1.25'.
-15 + 2x + x2 = 0

Move the constant term to the right:

Add '15' to each side of the equation.
-15 + 2x + 15 + x2 = 0 + 15

Reorder the terms:
-15 + 15 + 2x + x2 = 0 + 15

Combine like terms: -15 + 15 = 0
0 + 2x + x2 = 0 + 15
2x + x2 = 0 + 15

Combine like terms: 0 + 15 = 15
2x + x2 = 15

The x term is 2x.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2x + 1 + x2 = 15 + 1

Reorder the terms:
1 + 2x + x2 = 15 + 1

Combine like terms: 15 + 1 = 16
1 + 2x + x2 = 16

Factor a perfect square on the left side:
(x + 1)(x + 1) = 16

Calculate the square root of the right side: 4

Break this problem into two subproblems by setting 
(x + 1) equal to 4 and -4.

Subproblem 1

x + 1 = 4 Simplifying x + 1 = 4 Reorder the terms: 1 + x = 4 Solving 1 + x = 4 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = 4 + -1 Combine like terms: 1 + -1 = 0 0 + x = 4 + -1 x = 4 + -1 Combine like terms: 4 + -1 = 3 x = 3 Simplifying x = 3

Subproblem 2

x + 1 = -4 Simplifying x + 1 = -4 Reorder the terms: 1 + x = -4 Solving 1 + x = -4 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = -4 + -1 Combine like terms: 1 + -1 = 0 0 + x = -4 + -1 x = -4 + -1 Combine like terms: -4 + -1 = -5 x = -5 Simplifying x = -5

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3, -5}

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