4a^2-a-24=0

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Solution for 4a^2-a-24=0 equation:


Simplifying
4a2 + -1a + -24 = 0

Reorder the terms:
-24 + -1a + 4a2 = 0

Solving
-24 + -1a + 4a2 = 0

Solving for variable 'a'.

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

Divide each side by '4'.
-6 + -0.25a + a2 = 0

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + -0.25a + 6 + a2 = 0 + 6

Reorder the terms:
-6 + 6 + -0.25a + a2 = 0 + 6

Combine like terms: -6 + 6 = 0
0 + -0.25a + a2 = 0 + 6
-0.25a + a2 = 0 + 6

Combine like terms: 0 + 6 = 6
-0.25a + a2 = 6

The a term is -0.25a.  Take half its coefficient (-0.125).
Square it (0.015625) and add it to both sides.

Add '0.015625' to each side of the equation.
-0.25a + 0.015625 + a2 = 6 + 0.015625

Reorder the terms:
0.015625 + -0.25a + a2 = 6 + 0.015625

Combine like terms: 6 + 0.015625 = 6.015625
0.015625 + -0.25a + a2 = 6.015625

Factor a perfect square on the left side:
(a + -0.125)(a + -0.125) = 6.015625

Calculate the square root of the right side: 2.452677109

Break this problem into two subproblems by setting 
(a + -0.125) equal to 2.452677109 and -2.452677109.

Subproblem 1

a + -0.125 = 2.452677109 Simplifying a + -0.125 = 2.452677109 Reorder the terms: -0.125 + a = 2.452677109 Solving -0.125 + a = 2.452677109 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + a = 2.452677109 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + a = 2.452677109 + 0.125 a = 2.452677109 + 0.125 Combine like terms: 2.452677109 + 0.125 = 2.577677109 a = 2.577677109 Simplifying a = 2.577677109

Subproblem 2

a + -0.125 = -2.452677109 Simplifying a + -0.125 = -2.452677109 Reorder the terms: -0.125 + a = -2.452677109 Solving -0.125 + a = -2.452677109 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + a = -2.452677109 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + a = -2.452677109 + 0.125 a = -2.452677109 + 0.125 Combine like terms: -2.452677109 + 0.125 = -2.327677109 a = -2.327677109 Simplifying a = -2.327677109

Solution

The solution to the problem is based on the solutions from the subproblems. a = {2.577677109, -2.327677109}

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