a^2+3a-5=0

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Solution for a^2+3a-5=0 equation:


Simplifying
a2 + 3a + -5 = 0

Reorder the terms:
-5 + 3a + a2 = 0

Solving
-5 + 3a + a2 = 0

Solving for variable 'a'.

Begin completing the square.

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + 3a + 5 + a2 = 0 + 5

Reorder the terms:
-5 + 5 + 3a + a2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + 3a + a2 = 0 + 5
3a + a2 = 0 + 5

Combine like terms: 0 + 5 = 5
3a + a2 = 5

The a term is 3a.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3a + 2.25 + a2 = 5 + 2.25

Reorder the terms:
2.25 + 3a + a2 = 5 + 2.25

Combine like terms: 5 + 2.25 = 7.25
2.25 + 3a + a2 = 7.25

Factor a perfect square on the left side:
(a + 1.5)(a + 1.5) = 7.25

Calculate the square root of the right side: 2.692582404

Break this problem into two subproblems by setting 
(a + 1.5) equal to 2.692582404 and -2.692582404.

Subproblem 1

a + 1.5 = 2.692582404 Simplifying a + 1.5 = 2.692582404 Reorder the terms: 1.5 + a = 2.692582404 Solving 1.5 + a = 2.692582404 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + a = 2.692582404 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + a = 2.692582404 + -1.5 a = 2.692582404 + -1.5 Combine like terms: 2.692582404 + -1.5 = 1.192582404 a = 1.192582404 Simplifying a = 1.192582404

Subproblem 2

a + 1.5 = -2.692582404 Simplifying a + 1.5 = -2.692582404 Reorder the terms: 1.5 + a = -2.692582404 Solving 1.5 + a = -2.692582404 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + a = -2.692582404 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + a = -2.692582404 + -1.5 a = -2.692582404 + -1.5 Combine like terms: -2.692582404 + -1.5 = -4.192582404 a = -4.192582404 Simplifying a = -4.192582404

Solution

The solution to the problem is based on the solutions from the subproblems. a = {1.192582404, -4.192582404}

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