2a^2+a-90=0

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Solution for 2a^2+a-90=0 equation:


Simplifying
2a2 + a + -90 = 0

Reorder the terms:
-90 + a + 2a2 = 0

Solving
-90 + a + 2a2 = 0

Solving for variable 'a'.

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

Divide each side by '2'.
-45 + 0.5a + a2 = 0

Move the constant term to the right:

Add '45' to each side of the equation.
-45 + 0.5a + 45 + a2 = 0 + 45

Reorder the terms:
-45 + 45 + 0.5a + a2 = 0 + 45

Combine like terms: -45 + 45 = 0
0 + 0.5a + a2 = 0 + 45
0.5a + a2 = 0 + 45

Combine like terms: 0 + 45 = 45
0.5a + a2 = 45

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.
0.5a + 0.25 + a2 = 45 + 0.25

Reorder the terms:
0.25 + 0.5a + a2 = 45 + 0.25

Combine like terms: 45 + 0.25 = 45.25
0.25 + 0.5a + a2 = 45.25

Factor a perfect square on the left side:
(a + 0.5)(a + 0.5) = 45.25

Calculate the square root of the right side: 6.726812024

Break this problem into two subproblems by setting 
(a + 0.5) equal to 6.726812024 and -6.726812024.

Subproblem 1

a + 0.5 = 6.726812024 Simplifying a + 0.5 = 6.726812024 Reorder the terms: 0.5 + a = 6.726812024 Solving 0.5 + a = 6.726812024 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 = 6.726812024 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = 6.726812024 + -0.5 a = 6.726812024 + -0.5 Combine like terms: 6.726812024 + -0.5 = 6.226812024 a = 6.226812024 Simplifying a = 6.226812024

Subproblem 2

a + 0.5 = -6.726812024 Simplifying a + 0.5 = -6.726812024 Reorder the terms: 0.5 + a = -6.726812024 Solving 0.5 + a = -6.726812024 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 = -6.726812024 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = -6.726812024 + -0.5 a = -6.726812024 + -0.5 Combine like terms: -6.726812024 + -0.5 = -7.226812024 a = -7.226812024 Simplifying a = -7.226812024

Solution

The solution to the problem is based on the solutions from the subproblems. a = {6.226812024, -7.226812024}

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