5a^2+24a+5=0

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


Simplifying
5a2 + 24a + 5 = 0

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

Solving
5 + 24a + 5a2 = 0

Solving for variable 'a'.

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

Divide each side by '5'.
1 + 4.8a + a2 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 4.8a + -1 + a2 = 0 + -1

Reorder the terms:
1 + -1 + 4.8a + a2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 4.8a + a2 = 0 + -1
4.8a + a2 = 0 + -1

Combine like terms: 0 + -1 = -1
4.8a + a2 = -1

The a term is 4.8a.  Take half its coefficient (2.4).
Square it (5.76) and add it to both sides.

Add '5.76' to each side of the equation.
4.8a + 5.76 + a2 = -1 + 5.76

Reorder the terms:
5.76 + 4.8a + a2 = -1 + 5.76

Combine like terms: -1 + 5.76 = 4.76
5.76 + 4.8a + a2 = 4.76

Factor a perfect square on the left side:
(a + 2.4)(a + 2.4) = 4.76

Calculate the square root of the right side: 2.181742423

Break this problem into two subproblems by setting 
(a + 2.4) equal to 2.181742423 and -2.181742423.

Subproblem 1

a + 2.4 = 2.181742423 Simplifying a + 2.4 = 2.181742423 Reorder the terms: 2.4 + a = 2.181742423 Solving 2.4 + a = 2.181742423 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.4' to each side of the equation. 2.4 + -2.4 + a = 2.181742423 + -2.4 Combine like terms: 2.4 + -2.4 = 0.0 0.0 + a = 2.181742423 + -2.4 a = 2.181742423 + -2.4 Combine like terms: 2.181742423 + -2.4 = -0.218257577 a = -0.218257577 Simplifying a = -0.218257577

Subproblem 2

a + 2.4 = -2.181742423 Simplifying a + 2.4 = -2.181742423 Reorder the terms: 2.4 + a = -2.181742423 Solving 2.4 + a = -2.181742423 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.4' to each side of the equation. 2.4 + -2.4 + a = -2.181742423 + -2.4 Combine like terms: 2.4 + -2.4 = 0.0 0.0 + a = -2.181742423 + -2.4 a = -2.181742423 + -2.4 Combine like terms: -2.181742423 + -2.4 = -4.581742423 a = -4.581742423 Simplifying a = -4.581742423

Solution

The solution to the problem is based on the solutions from the subproblems. a = {-0.218257577, -4.581742423}

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