a^2+a=3

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Solution for a^2+a=3 equation:


Simplifying
a2 + a = 3

Reorder the terms:
a + a2 = 3

Solving
a + a2 = 3

Solving for variable 'a'.

Reorder the terms:
-3 + a + a2 = 3 + -3

Combine like terms: 3 + -3 = 0
-3 + a + a2 = 0

Begin completing the square.

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + a + 3 + a2 = 0 + 3

Reorder the terms:
-3 + 3 + a + a2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + a + a2 = 0 + 3
a + a2 = 0 + 3

Combine like terms: 0 + 3 = 3
a + a2 = 3

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 = 3 + 0.25

Reorder the terms:
0.25 + a + a2 = 3 + 0.25

Combine like terms: 3 + 0.25 = 3.25
0.25 + a + a2 = 3.25

Factor a perfect square on the left side:
(a + 0.5)(a + 0.5) = 3.25

Calculate the square root of the right side: 1.802775638

Break this problem into two subproblems by setting 
(a + 0.5) equal to 1.802775638 and -1.802775638.

Subproblem 1

a + 0.5 = 1.802775638 Simplifying a + 0.5 = 1.802775638 Reorder the terms: 0.5 + a = 1.802775638 Solving 0.5 + a = 1.802775638 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 = 1.802775638 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = 1.802775638 + -0.5 a = 1.802775638 + -0.5 Combine like terms: 1.802775638 + -0.5 = 1.302775638 a = 1.302775638 Simplifying a = 1.302775638

Subproblem 2

a + 0.5 = -1.802775638 Simplifying a + 0.5 = -1.802775638 Reorder the terms: 0.5 + a = -1.802775638 Solving 0.5 + a = -1.802775638 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 = -1.802775638 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = -1.802775638 + -0.5 a = -1.802775638 + -0.5 Combine like terms: -1.802775638 + -0.5 = -2.302775638 a = -2.302775638 Simplifying a = -2.302775638

Solution

The solution to the problem is based on the solutions from the subproblems. a = {1.302775638, -2.302775638}

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