8a^2=-71a-90

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Solution for 8a^2=-71a-90 equation:


Simplifying
8a2 = -71a + -90

Reorder the terms:
8a2 = -90 + -71a

Solving
8a2 = -90 + -71a

Solving for variable 'a'.

Reorder the terms:
90 + 71a + 8a2 = -90 + -71a + 90 + 71a

Reorder the terms:
90 + 71a + 8a2 = -90 + 90 + -71a + 71a

Combine like terms: -90 + 90 = 0
90 + 71a + 8a2 = 0 + -71a + 71a
90 + 71a + 8a2 = -71a + 71a

Combine like terms: -71a + 71a = 0
90 + 71a + 8a2 = 0

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

Divide each side by '8'.
11.25 + 8.875a + a2 = 0

Move the constant term to the right:

Add '-11.25' to each side of the equation.
11.25 + 8.875a + -11.25 + a2 = 0 + -11.25

Reorder the terms:
11.25 + -11.25 + 8.875a + a2 = 0 + -11.25

Combine like terms: 11.25 + -11.25 = 0.00
0.00 + 8.875a + a2 = 0 + -11.25
8.875a + a2 = 0 + -11.25

Combine like terms: 0 + -11.25 = -11.25
8.875a + a2 = -11.25

The a term is 8.875a.  Take half its coefficient (4.4375).
Square it (19.69140625) and add it to both sides.

Add '19.69140625' to each side of the equation.
8.875a + 19.69140625 + a2 = -11.25 + 19.69140625

Reorder the terms:
19.69140625 + 8.875a + a2 = -11.25 + 19.69140625

Combine like terms: -11.25 + 19.69140625 = 8.44140625
19.69140625 + 8.875a + a2 = 8.44140625

Factor a perfect square on the left side:
(a + 4.4375)(a + 4.4375) = 8.44140625

Calculate the square root of the right side: 2.905409825

Break this problem into two subproblems by setting 
(a + 4.4375) equal to 2.905409825 and -2.905409825.

Subproblem 1

a + 4.4375 = 2.905409825 Simplifying a + 4.4375 = 2.905409825 Reorder the terms: 4.4375 + a = 2.905409825 Solving 4.4375 + a = 2.905409825 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-4.4375' to each side of the equation. 4.4375 + -4.4375 + a = 2.905409825 + -4.4375 Combine like terms: 4.4375 + -4.4375 = 0.0000 0.0000 + a = 2.905409825 + -4.4375 a = 2.905409825 + -4.4375 Combine like terms: 2.905409825 + -4.4375 = -1.532090175 a = -1.532090175 Simplifying a = -1.532090175

Subproblem 2

a + 4.4375 = -2.905409825 Simplifying a + 4.4375 = -2.905409825 Reorder the terms: 4.4375 + a = -2.905409825 Solving 4.4375 + a = -2.905409825 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-4.4375' to each side of the equation. 4.4375 + -4.4375 + a = -2.905409825 + -4.4375 Combine like terms: 4.4375 + -4.4375 = 0.0000 0.0000 + a = -2.905409825 + -4.4375 a = -2.905409825 + -4.4375 Combine like terms: -2.905409825 + -4.4375 = -7.342909825 a = -7.342909825 Simplifying a = -7.342909825

Solution

The solution to the problem is based on the solutions from the subproblems. a = {-1.532090175, -7.342909825}

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