0=a^2-24a-48

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Solution for 0=a^2-24a-48 equation:


Simplifying
0 = a2 + -24a + -48

Reorder the terms:
0 = -48 + -24a + a2

Solving
0 = -48 + -24a + a2

Solving for variable 'a'.

Combine like terms: 0 + 48 = 48
48 + 24a + -1a2 = -48 + -24a + a2 + 48 + 24a + -1a2

Reorder the terms:
48 + 24a + -1a2 = -48 + 48 + -24a + 24a + a2 + -1a2

Combine like terms: -48 + 48 = 0
48 + 24a + -1a2 = 0 + -24a + 24a + a2 + -1a2
48 + 24a + -1a2 = -24a + 24a + a2 + -1a2

Combine like terms: -24a + 24a = 0
48 + 24a + -1a2 = 0 + a2 + -1a2
48 + 24a + -1a2 = a2 + -1a2

Combine like terms: a2 + -1a2 = 0
48 + 24a + -1a2 = 0

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

Divide each side by '-1'.
-48 + -24a + a2 = 0

Move the constant term to the right:

Add '48' to each side of the equation.
-48 + -24a + 48 + a2 = 0 + 48

Reorder the terms:
-48 + 48 + -24a + a2 = 0 + 48

Combine like terms: -48 + 48 = 0
0 + -24a + a2 = 0 + 48
-24a + a2 = 0 + 48

Combine like terms: 0 + 48 = 48
-24a + a2 = 48

The a term is -24a.  Take half its coefficient (-12).
Square it (144) and add it to both sides.

Add '144' to each side of the equation.
-24a + 144 + a2 = 48 + 144

Reorder the terms:
144 + -24a + a2 = 48 + 144

Combine like terms: 48 + 144 = 192
144 + -24a + a2 = 192

Factor a perfect square on the left side:
(a + -12)(a + -12) = 192

Calculate the square root of the right side: 13.856406461

Break this problem into two subproblems by setting 
(a + -12) equal to 13.856406461 and -13.856406461.

Subproblem 1

a + -12 = 13.856406461 Simplifying a + -12 = 13.856406461 Reorder the terms: -12 + a = 13.856406461 Solving -12 + a = 13.856406461 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '12' to each side of the equation. -12 + 12 + a = 13.856406461 + 12 Combine like terms: -12 + 12 = 0 0 + a = 13.856406461 + 12 a = 13.856406461 + 12 Combine like terms: 13.856406461 + 12 = 25.856406461 a = 25.856406461 Simplifying a = 25.856406461

Subproblem 2

a + -12 = -13.856406461 Simplifying a + -12 = -13.856406461 Reorder the terms: -12 + a = -13.856406461 Solving -12 + a = -13.856406461 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '12' to each side of the equation. -12 + 12 + a = -13.856406461 + 12 Combine like terms: -12 + 12 = 0 0 + a = -13.856406461 + 12 a = -13.856406461 + 12 Combine like terms: -13.856406461 + 12 = -1.856406461 a = -1.856406461 Simplifying a = -1.856406461

Solution

The solution to the problem is based on the solutions from the subproblems. a = {25.856406461, -1.856406461}

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