0=a^2+3a-90

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Solution for 0=a^2+3a-90 equation:


Simplifying
0 = a2 + 3a + -90

Reorder the terms:
0 = -90 + 3a + a2

Solving
0 = -90 + 3a + a2

Solving for variable 'a'.

Combine like terms: 0 + 90 = 90
90 + -3a + -1a2 = -90 + 3a + a2 + 90 + -3a + -1a2

Reorder the terms:
90 + -3a + -1a2 = -90 + 90 + 3a + -3a + a2 + -1a2

Combine like terms: -90 + 90 = 0
90 + -3a + -1a2 = 0 + 3a + -3a + a2 + -1a2
90 + -3a + -1a2 = 3a + -3a + a2 + -1a2

Combine like terms: 3a + -3a = 0
90 + -3a + -1a2 = 0 + a2 + -1a2
90 + -3a + -1a2 = a2 + -1a2

Combine like terms: a2 + -1a2 = 0
90 + -3a + -1a2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-90 + 3a + a2 = 0

Move the constant term to the right:

Add '90' to each side of the equation.
-90 + 3a + 90 + a2 = 0 + 90

Reorder the terms:
-90 + 90 + 3a + a2 = 0 + 90

Combine like terms: -90 + 90 = 0
0 + 3a + a2 = 0 + 90
3a + a2 = 0 + 90

Combine like terms: 0 + 90 = 90
3a + a2 = 90

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 = 90 + 2.25

Reorder the terms:
2.25 + 3a + a2 = 90 + 2.25

Combine like terms: 90 + 2.25 = 92.25
2.25 + 3a + a2 = 92.25

Factor a perfect square on the left side:
(a + 1.5)(a + 1.5) = 92.25

Calculate the square root of the right side: 9.604686356

Break this problem into two subproblems by setting 
(a + 1.5) equal to 9.604686356 and -9.604686356.

Subproblem 1

a + 1.5 = 9.604686356 Simplifying a + 1.5 = 9.604686356 Reorder the terms: 1.5 + a = 9.604686356 Solving 1.5 + a = 9.604686356 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 = 9.604686356 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + a = 9.604686356 + -1.5 a = 9.604686356 + -1.5 Combine like terms: 9.604686356 + -1.5 = 8.104686356 a = 8.104686356 Simplifying a = 8.104686356

Subproblem 2

a + 1.5 = -9.604686356 Simplifying a + 1.5 = -9.604686356 Reorder the terms: 1.5 + a = -9.604686356 Solving 1.5 + a = -9.604686356 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 = -9.604686356 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + a = -9.604686356 + -1.5 a = -9.604686356 + -1.5 Combine like terms: -9.604686356 + -1.5 = -11.104686356 a = -11.104686356 Simplifying a = -11.104686356

Solution

The solution to the problem is based on the solutions from the subproblems. a = {8.104686356, -11.104686356}

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