8a^2+28a-31=0

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Solution for 8a^2+28a-31=0 equation:


Simplifying
8a2 + 28a + -31 = 0

Reorder the terms:
-31 + 28a + 8a2 = 0

Solving
-31 + 28a + 8a2 = 0

Solving for variable 'a'.

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

Divide each side by '8'.
-3.875 + 3.5a + a2 = 0

Move the constant term to the right:

Add '3.875' to each side of the equation.
-3.875 + 3.5a + 3.875 + a2 = 0 + 3.875

Reorder the terms:
-3.875 + 3.875 + 3.5a + a2 = 0 + 3.875

Combine like terms: -3.875 + 3.875 = 0.000
0.000 + 3.5a + a2 = 0 + 3.875
3.5a + a2 = 0 + 3.875

Combine like terms: 0 + 3.875 = 3.875
3.5a + a2 = 3.875

The a term is 3.5a.  Take half its coefficient (1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
3.5a + 3.0625 + a2 = 3.875 + 3.0625

Reorder the terms:
3.0625 + 3.5a + a2 = 3.875 + 3.0625

Combine like terms: 3.875 + 3.0625 = 6.9375
3.0625 + 3.5a + a2 = 6.9375

Factor a perfect square on the left side:
(a + 1.75)(a + 1.75) = 6.9375

Calculate the square root of the right side: 2.633913438

Break this problem into two subproblems by setting 
(a + 1.75) equal to 2.633913438 and -2.633913438.

Subproblem 1

a + 1.75 = 2.633913438 Simplifying a + 1.75 = 2.633913438 Reorder the terms: 1.75 + a = 2.633913438 Solving 1.75 + a = 2.633913438 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + a = 2.633913438 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + a = 2.633913438 + -1.75 a = 2.633913438 + -1.75 Combine like terms: 2.633913438 + -1.75 = 0.883913438 a = 0.883913438 Simplifying a = 0.883913438

Subproblem 2

a + 1.75 = -2.633913438 Simplifying a + 1.75 = -2.633913438 Reorder the terms: 1.75 + a = -2.633913438 Solving 1.75 + a = -2.633913438 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + a = -2.633913438 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + a = -2.633913438 + -1.75 a = -2.633913438 + -1.75 Combine like terms: -2.633913438 + -1.75 = -4.383913438 a = -4.383913438 Simplifying a = -4.383913438

Solution

The solution to the problem is based on the solutions from the subproblems. a = {0.883913438, -4.383913438}

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