10r^2=-9r+8

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Solution for 10r^2=-9r+8 equation:


Simplifying
10r2 = -9r + 8

Reorder the terms:
10r2 = 8 + -9r

Solving
10r2 = 8 + -9r

Solving for variable 'r'.

Reorder the terms:
-8 + 9r + 10r2 = 8 + -9r + -8 + 9r

Reorder the terms:
-8 + 9r + 10r2 = 8 + -8 + -9r + 9r

Combine like terms: 8 + -8 = 0
-8 + 9r + 10r2 = 0 + -9r + 9r
-8 + 9r + 10r2 = -9r + 9r

Combine like terms: -9r + 9r = 0
-8 + 9r + 10r2 = 0

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

Divide each side by '10'.
-0.8 + 0.9r + r2 = 0

Move the constant term to the right:

Add '0.8' to each side of the equation.
-0.8 + 0.9r + 0.8 + r2 = 0 + 0.8

Reorder the terms:
-0.8 + 0.8 + 0.9r + r2 = 0 + 0.8

Combine like terms: -0.8 + 0.8 = 0.0
0.0 + 0.9r + r2 = 0 + 0.8
0.9r + r2 = 0 + 0.8

Combine like terms: 0 + 0.8 = 0.8
0.9r + r2 = 0.8

The r term is 0.9r.  Take half its coefficient (0.45).
Square it (0.2025) and add it to both sides.

Add '0.2025' to each side of the equation.
0.9r + 0.2025 + r2 = 0.8 + 0.2025

Reorder the terms:
0.2025 + 0.9r + r2 = 0.8 + 0.2025

Combine like terms: 0.8 + 0.2025 = 1.0025
0.2025 + 0.9r + r2 = 1.0025

Factor a perfect square on the left side:
(r + 0.45)(r + 0.45) = 1.0025

Calculate the square root of the right side: 1.00124922

Break this problem into two subproblems by setting 
(r + 0.45) equal to 1.00124922 and -1.00124922.

Subproblem 1

r + 0.45 = 1.00124922 Simplifying r + 0.45 = 1.00124922 Reorder the terms: 0.45 + r = 1.00124922 Solving 0.45 + r = 1.00124922 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.45' to each side of the equation. 0.45 + -0.45 + r = 1.00124922 + -0.45 Combine like terms: 0.45 + -0.45 = 0.00 0.00 + r = 1.00124922 + -0.45 r = 1.00124922 + -0.45 Combine like terms: 1.00124922 + -0.45 = 0.55124922 r = 0.55124922 Simplifying r = 0.55124922

Subproblem 2

r + 0.45 = -1.00124922 Simplifying r + 0.45 = -1.00124922 Reorder the terms: 0.45 + r = -1.00124922 Solving 0.45 + r = -1.00124922 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.45' to each side of the equation. 0.45 + -0.45 + r = -1.00124922 + -0.45 Combine like terms: 0.45 + -0.45 = 0.00 0.00 + r = -1.00124922 + -0.45 r = -1.00124922 + -0.45 Combine like terms: -1.00124922 + -0.45 = -1.45124922 r = -1.45124922 Simplifying r = -1.45124922

Solution

The solution to the problem is based on the solutions from the subproblems. r = {0.55124922, -1.45124922}

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