4a^2+5a-24=0

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Solution for 4a^2+5a-24=0 equation:


Simplifying
4a2 + 5a + -24 = 0

Reorder the terms:
-24 + 5a + 4a2 = 0

Solving
-24 + 5a + 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 + 1.25a + a2 = 0

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + 1.25a + 6 + a2 = 0 + 6

Reorder the terms:
-6 + 6 + 1.25a + a2 = 0 + 6

Combine like terms: -6 + 6 = 0
0 + 1.25a + a2 = 0 + 6
1.25a + a2 = 0 + 6

Combine like terms: 0 + 6 = 6
1.25a + a2 = 6

The a term is 1.25a.  Take half its coefficient (0.625).
Square it (0.390625) and add it to both sides.

Add '0.390625' to each side of the equation.
1.25a + 0.390625 + a2 = 6 + 0.390625

Reorder the terms:
0.390625 + 1.25a + a2 = 6 + 0.390625

Combine like terms: 6 + 0.390625 = 6.390625
0.390625 + 1.25a + a2 = 6.390625

Factor a perfect square on the left side:
(a + 0.625)(a + 0.625) = 6.390625

Calculate the square root of the right side: 2.527968552

Break this problem into two subproblems by setting 
(a + 0.625) equal to 2.527968552 and -2.527968552.

Subproblem 1

a + 0.625 = 2.527968552 Simplifying a + 0.625 = 2.527968552 Reorder the terms: 0.625 + a = 2.527968552 Solving 0.625 + a = 2.527968552 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + a = 2.527968552 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + a = 2.527968552 + -0.625 a = 2.527968552 + -0.625 Combine like terms: 2.527968552 + -0.625 = 1.902968552 a = 1.902968552 Simplifying a = 1.902968552

Subproblem 2

a + 0.625 = -2.527968552 Simplifying a + 0.625 = -2.527968552 Reorder the terms: 0.625 + a = -2.527968552 Solving 0.625 + a = -2.527968552 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + a = -2.527968552 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + a = -2.527968552 + -0.625 a = -2.527968552 + -0.625 Combine like terms: -2.527968552 + -0.625 = -3.152968552 a = -3.152968552 Simplifying a = -3.152968552

Solution

The solution to the problem is based on the solutions from the subproblems. a = {1.902968552, -3.152968552}

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