90a^2=-103a-24

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Solution for 90a^2=-103a-24 equation:


Simplifying
90a2 = -103a + -24

Reorder the terms:
90a2 = -24 + -103a

Solving
90a2 = -24 + -103a

Solving for variable 'a'.

Reorder the terms:
24 + 103a + 90a2 = -24 + -103a + 24 + 103a

Reorder the terms:
24 + 103a + 90a2 = -24 + 24 + -103a + 103a

Combine like terms: -24 + 24 = 0
24 + 103a + 90a2 = 0 + -103a + 103a
24 + 103a + 90a2 = -103a + 103a

Combine like terms: -103a + 103a = 0
24 + 103a + 90a2 = 0

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

Divide each side by '90'.
0.2666666667 + 1.144444444a + a2 = 0

Move the constant term to the right:

Add '-0.2666666667' to each side of the equation.
0.2666666667 + 1.144444444a + -0.2666666667 + a2 = 0 + -0.2666666667

Reorder the terms:
0.2666666667 + -0.2666666667 + 1.144444444a + a2 = 0 + -0.2666666667

Combine like terms: 0.2666666667 + -0.2666666667 = 0.0000000000
0.0000000000 + 1.144444444a + a2 = 0 + -0.2666666667
1.144444444a + a2 = 0 + -0.2666666667

Combine like terms: 0 + -0.2666666667 = -0.2666666667
1.144444444a + a2 = -0.2666666667

The a term is 1.144444444a.  Take half its coefficient (0.572222222).
Square it (0.3274382714) and add it to both sides.

Add '0.3274382714' to each side of the equation.
1.144444444a + 0.3274382714 + a2 = -0.2666666667 + 0.3274382714

Reorder the terms:
0.3274382714 + 1.144444444a + a2 = -0.2666666667 + 0.3274382714

Combine like terms: -0.2666666667 + 0.3274382714 = 0.0607716047
0.3274382714 + 1.144444444a + a2 = 0.0607716047

Factor a perfect square on the left side:
(a + 0.572222222)(a + 0.572222222) = 0.0607716047

Calculate the square root of the right side: 0.246518974

Break this problem into two subproblems by setting 
(a + 0.572222222) equal to 0.246518974 and -0.246518974.

Subproblem 1

a + 0.572222222 = 0.246518974 Simplifying a + 0.572222222 = 0.246518974 Reorder the terms: 0.572222222 + a = 0.246518974 Solving 0.572222222 + a = 0.246518974 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.572222222' to each side of the equation. 0.572222222 + -0.572222222 + a = 0.246518974 + -0.572222222 Combine like terms: 0.572222222 + -0.572222222 = 0.000000000 0.000000000 + a = 0.246518974 + -0.572222222 a = 0.246518974 + -0.572222222 Combine like terms: 0.246518974 + -0.572222222 = -0.325703248 a = -0.325703248 Simplifying a = -0.325703248

Subproblem 2

a + 0.572222222 = -0.246518974 Simplifying a + 0.572222222 = -0.246518974 Reorder the terms: 0.572222222 + a = -0.246518974 Solving 0.572222222 + a = -0.246518974 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.572222222' to each side of the equation. 0.572222222 + -0.572222222 + a = -0.246518974 + -0.572222222 Combine like terms: 0.572222222 + -0.572222222 = 0.000000000 0.000000000 + a = -0.246518974 + -0.572222222 a = -0.246518974 + -0.572222222 Combine like terms: -0.246518974 + -0.572222222 = -0.818741196 a = -0.818741196 Simplifying a = -0.818741196

Solution

The solution to the problem is based on the solutions from the subproblems. a = {-0.325703248, -0.818741196}

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