a^2+a-224=0

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Solution for a^2+a-224=0 equation:


Simplifying
a2 + a + -224 = 0

Reorder the terms:
-224 + a + a2 = 0

Solving
-224 + a + a2 = 0

Solving for variable 'a'.

Begin completing the square.

Move the constant term to the right:

Add '224' to each side of the equation.
-224 + a + 224 + a2 = 0 + 224

Reorder the terms:
-224 + 224 + a + a2 = 0 + 224

Combine like terms: -224 + 224 = 0
0 + a + a2 = 0 + 224
a + a2 = 0 + 224

Combine like terms: 0 + 224 = 224
a + a2 = 224

The a term is a.  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.
a + 0.25 + a2 = 224 + 0.25

Reorder the terms:
0.25 + a + a2 = 224 + 0.25

Combine like terms: 224 + 0.25 = 224.25
0.25 + a + a2 = 224.25

Factor a perfect square on the left side:
(a + 0.5)(a + 0.5) = 224.25

Calculate the square root of the right side: 14.974979132

Break this problem into two subproblems by setting 
(a + 0.5) equal to 14.974979132 and -14.974979132.

Subproblem 1

a + 0.5 = 14.974979132 Simplifying a + 0.5 = 14.974979132 Reorder the terms: 0.5 + a = 14.974979132 Solving 0.5 + a = 14.974979132 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + a = 14.974979132 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = 14.974979132 + -0.5 a = 14.974979132 + -0.5 Combine like terms: 14.974979132 + -0.5 = 14.474979132 a = 14.474979132 Simplifying a = 14.474979132

Subproblem 2

a + 0.5 = -14.974979132 Simplifying a + 0.5 = -14.974979132 Reorder the terms: 0.5 + a = -14.974979132 Solving 0.5 + a = -14.974979132 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + a = -14.974979132 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = -14.974979132 + -0.5 a = -14.974979132 + -0.5 Combine like terms: -14.974979132 + -0.5 = -15.474979132 a = -15.474979132 Simplifying a = -15.474979132

Solution

The solution to the problem is based on the solutions from the subproblems. a = {14.474979132, -15.474979132}

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