7r=-3r^2+9

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Solution for 7r=-3r^2+9 equation:


Simplifying
7r = -3r2 + 9

Reorder the terms:
7r = 9 + -3r2

Solving
7r = 9 + -3r2

Solving for variable 'r'.

Reorder the terms:
-9 + 7r + 3r2 = 9 + -3r2 + -9 + 3r2

Reorder the terms:
-9 + 7r + 3r2 = 9 + -9 + -3r2 + 3r2

Combine like terms: 9 + -9 = 0
-9 + 7r + 3r2 = 0 + -3r2 + 3r2
-9 + 7r + 3r2 = -3r2 + 3r2

Combine like terms: -3r2 + 3r2 = 0
-9 + 7r + 3r2 = 0

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

Divide each side by '3'.
-3 + 2.333333333r + r2 = 0

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + 2.333333333r + 3 + r2 = 0 + 3

Reorder the terms:
-3 + 3 + 2.333333333r + r2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + 2.333333333r + r2 = 0 + 3
2.333333333r + r2 = 0 + 3

Combine like terms: 0 + 3 = 3
2.333333333r + r2 = 3

The r term is 2.333333333r.  Take half its coefficient (1.166666667).
Square it (1.361111112) and add it to both sides.

Add '1.361111112' to each side of the equation.
2.333333333r + 1.361111112 + r2 = 3 + 1.361111112

Reorder the terms:
1.361111112 + 2.333333333r + r2 = 3 + 1.361111112

Combine like terms: 3 + 1.361111112 = 4.361111112
1.361111112 + 2.333333333r + r2 = 4.361111112

Factor a perfect square on the left side:
(r + 1.166666667)(r + 1.166666667) = 4.361111112

Calculate the square root of the right side: 2.088327348

Break this problem into two subproblems by setting 
(r + 1.166666667) equal to 2.088327348 and -2.088327348.

Subproblem 1

r + 1.166666667 = 2.088327348 Simplifying r + 1.166666667 = 2.088327348 Reorder the terms: 1.166666667 + r = 2.088327348 Solving 1.166666667 + r = 2.088327348 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.166666667' to each side of the equation. 1.166666667 + -1.166666667 + r = 2.088327348 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + r = 2.088327348 + -1.166666667 r = 2.088327348 + -1.166666667 Combine like terms: 2.088327348 + -1.166666667 = 0.921660681 r = 0.921660681 Simplifying r = 0.921660681

Subproblem 2

r + 1.166666667 = -2.088327348 Simplifying r + 1.166666667 = -2.088327348 Reorder the terms: 1.166666667 + r = -2.088327348 Solving 1.166666667 + r = -2.088327348 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.166666667' to each side of the equation. 1.166666667 + -1.166666667 + r = -2.088327348 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + r = -2.088327348 + -1.166666667 r = -2.088327348 + -1.166666667 Combine like terms: -2.088327348 + -1.166666667 = -3.254994015 r = -3.254994015 Simplifying r = -3.254994015

Solution

The solution to the problem is based on the solutions from the subproblems. r = {0.921660681, -3.254994015}

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