a^2+a-21=0

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Solution for a^2+a-21=0 equation:


Simplifying
a2 + a + -21 = 0

Reorder the terms:
-21 + a + a2 = 0

Solving
-21 + a + a2 = 0

Solving for variable 'a'.

Begin completing the square.

Move the constant term to the right:

Add '21' to each side of the equation.
-21 + a + 21 + a2 = 0 + 21

Reorder the terms:
-21 + 21 + a + a2 = 0 + 21

Combine like terms: -21 + 21 = 0
0 + a + a2 = 0 + 21
a + a2 = 0 + 21

Combine like terms: 0 + 21 = 21
a + a2 = 21

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 = 21 + 0.25

Reorder the terms:
0.25 + a + a2 = 21 + 0.25

Combine like terms: 21 + 0.25 = 21.25
0.25 + a + a2 = 21.25

Factor a perfect square on the left side:
(a + 0.5)(a + 0.5) = 21.25

Calculate the square root of the right side: 4.609772229

Break this problem into two subproblems by setting 
(a + 0.5) equal to 4.609772229 and -4.609772229.

Subproblem 1

a + 0.5 = 4.609772229 Simplifying a + 0.5 = 4.609772229 Reorder the terms: 0.5 + a = 4.609772229 Solving 0.5 + a = 4.609772229 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 = 4.609772229 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = 4.609772229 + -0.5 a = 4.609772229 + -0.5 Combine like terms: 4.609772229 + -0.5 = 4.109772229 a = 4.109772229 Simplifying a = 4.109772229

Subproblem 2

a + 0.5 = -4.609772229 Simplifying a + 0.5 = -4.609772229 Reorder the terms: 0.5 + a = -4.609772229 Solving 0.5 + a = -4.609772229 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 = -4.609772229 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = -4.609772229 + -0.5 a = -4.609772229 + -0.5 Combine like terms: -4.609772229 + -0.5 = -5.109772229 a = -5.109772229 Simplifying a = -5.109772229

Solution

The solution to the problem is based on the solutions from the subproblems. a = {4.109772229, -5.109772229}

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