5r^2=10-9r

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Solution for 5r^2=10-9r equation:


Simplifying
5r2 = 10 + -9r

Solving
5r2 = 10 + -9r

Solving for variable 'r'.

Reorder the terms:
-10 + 9r + 5r2 = 10 + -9r + -10 + 9r

Reorder the terms:
-10 + 9r + 5r2 = 10 + -10 + -9r + 9r

Combine like terms: 10 + -10 = 0
-10 + 9r + 5r2 = 0 + -9r + 9r
-10 + 9r + 5r2 = -9r + 9r

Combine like terms: -9r + 9r = 0
-10 + 9r + 5r2 = 0

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

Divide each side by '5'.
-2 + 1.8r + r2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + 1.8r + 2 + r2 = 0 + 2

Reorder the terms:
-2 + 2 + 1.8r + r2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + 1.8r + r2 = 0 + 2
1.8r + r2 = 0 + 2

Combine like terms: 0 + 2 = 2
1.8r + r2 = 2

The r term is 1.8r.  Take half its coefficient (0.9).
Square it (0.81) and add it to both sides.

Add '0.81' to each side of the equation.
1.8r + 0.81 + r2 = 2 + 0.81

Reorder the terms:
0.81 + 1.8r + r2 = 2 + 0.81

Combine like terms: 2 + 0.81 = 2.81
0.81 + 1.8r + r2 = 2.81

Factor a perfect square on the left side:
(r + 0.9)(r + 0.9) = 2.81

Calculate the square root of the right side: 1.676305461

Break this problem into two subproblems by setting 
(r + 0.9) equal to 1.676305461 and -1.676305461.

Subproblem 1

r + 0.9 = 1.676305461 Simplifying r + 0.9 = 1.676305461 Reorder the terms: 0.9 + r = 1.676305461 Solving 0.9 + r = 1.676305461 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.9' to each side of the equation. 0.9 + -0.9 + r = 1.676305461 + -0.9 Combine like terms: 0.9 + -0.9 = 0.0 0.0 + r = 1.676305461 + -0.9 r = 1.676305461 + -0.9 Combine like terms: 1.676305461 + -0.9 = 0.776305461 r = 0.776305461 Simplifying r = 0.776305461

Subproblem 2

r + 0.9 = -1.676305461 Simplifying r + 0.9 = -1.676305461 Reorder the terms: 0.9 + r = -1.676305461 Solving 0.9 + r = -1.676305461 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.9' to each side of the equation. 0.9 + -0.9 + r = -1.676305461 + -0.9 Combine like terms: 0.9 + -0.9 = 0.0 0.0 + r = -1.676305461 + -0.9 r = -1.676305461 + -0.9 Combine like terms: -1.676305461 + -0.9 = -2.576305461 r = -2.576305461 Simplifying r = -2.576305461

Solution

The solution to the problem is based on the solutions from the subproblems. r = {0.776305461, -2.576305461}

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