5a^2+8a-23=0

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Solution for 5a^2+8a-23=0 equation:


Simplifying
5a2 + 8a + -23 = 0

Reorder the terms:
-23 + 8a + 5a2 = 0

Solving
-23 + 8a + 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'.
-4.6 + 1.6a + a2 = 0

Move the constant term to the right:

Add '4.6' to each side of the equation.
-4.6 + 1.6a + 4.6 + a2 = 0 + 4.6

Reorder the terms:
-4.6 + 4.6 + 1.6a + a2 = 0 + 4.6

Combine like terms: -4.6 + 4.6 = 0.0
0.0 + 1.6a + a2 = 0 + 4.6
1.6a + a2 = 0 + 4.6

Combine like terms: 0 + 4.6 = 4.6
1.6a + a2 = 4.6

The a term is 1.6a.  Take half its coefficient (0.8).
Square it (0.64) and add it to both sides.

Add '0.64' to each side of the equation.
1.6a + 0.64 + a2 = 4.6 + 0.64

Reorder the terms:
0.64 + 1.6a + a2 = 4.6 + 0.64

Combine like terms: 4.6 + 0.64 = 5.24
0.64 + 1.6a + a2 = 5.24

Factor a perfect square on the left side:
(a + 0.8)(a + 0.8) = 5.24

Calculate the square root of the right side: 2.289104628

Break this problem into two subproblems by setting 
(a + 0.8) equal to 2.289104628 and -2.289104628.

Subproblem 1

a + 0.8 = 2.289104628 Simplifying a + 0.8 = 2.289104628 Reorder the terms: 0.8 + a = 2.289104628 Solving 0.8 + a = 2.289104628 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.8' to each side of the equation. 0.8 + -0.8 + a = 2.289104628 + -0.8 Combine like terms: 0.8 + -0.8 = 0.0 0.0 + a = 2.289104628 + -0.8 a = 2.289104628 + -0.8 Combine like terms: 2.289104628 + -0.8 = 1.489104628 a = 1.489104628 Simplifying a = 1.489104628

Subproblem 2

a + 0.8 = -2.289104628 Simplifying a + 0.8 = -2.289104628 Reorder the terms: 0.8 + a = -2.289104628 Solving 0.8 + a = -2.289104628 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.8' to each side of the equation. 0.8 + -0.8 + a = -2.289104628 + -0.8 Combine like terms: 0.8 + -0.8 = 0.0 0.0 + a = -2.289104628 + -0.8 a = -2.289104628 + -0.8 Combine like terms: -2.289104628 + -0.8 = -3.089104628 a = -3.089104628 Simplifying a = -3.089104628

Solution

The solution to the problem is based on the solutions from the subproblems. a = {1.489104628, -3.089104628}

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