3r^2+10r-21=3

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Solution for 3r^2+10r-21=3 equation:


Simplifying
3r2 + 10r + -21 = 3

Reorder the terms:
-21 + 10r + 3r2 = 3

Solving
-21 + 10r + 3r2 = 3

Solving for variable 'r'.

Reorder the terms:
-21 + -3 + 10r + 3r2 = 3 + -3

Combine like terms: -21 + -3 = -24
-24 + 10r + 3r2 = 3 + -3

Combine like terms: 3 + -3 = 0
-24 + 10r + 3r2 = 0

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

Divide each side by '3'.
-8 + 3.333333333r + r2 = 0

Move the constant term to the right:

Add '8' to each side of the equation.
-8 + 3.333333333r + 8 + r2 = 0 + 8

Reorder the terms:
-8 + 8 + 3.333333333r + r2 = 0 + 8

Combine like terms: -8 + 8 = 0
0 + 3.333333333r + r2 = 0 + 8
3.333333333r + r2 = 0 + 8

Combine like terms: 0 + 8 = 8
3.333333333r + r2 = 8

The r term is 3.333333333r.  Take half its coefficient (1.666666667).
Square it (2.777777779) and add it to both sides.

Add '2.777777779' to each side of the equation.
3.333333333r + 2.777777779 + r2 = 8 + 2.777777779

Reorder the terms:
2.777777779 + 3.333333333r + r2 = 8 + 2.777777779

Combine like terms: 8 + 2.777777779 = 10.777777779
2.777777779 + 3.333333333r + r2 = 10.777777779

Factor a perfect square on the left side:
(r + 1.666666667)(r + 1.666666667) = 10.777777779

Calculate the square root of the right side: 3.282952601

Break this problem into two subproblems by setting 
(r + 1.666666667) equal to 3.282952601 and -3.282952601.

Subproblem 1

r + 1.666666667 = 3.282952601 Simplifying r + 1.666666667 = 3.282952601 Reorder the terms: 1.666666667 + r = 3.282952601 Solving 1.666666667 + r = 3.282952601 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.666666667' to each side of the equation. 1.666666667 + -1.666666667 + r = 3.282952601 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + r = 3.282952601 + -1.666666667 r = 3.282952601 + -1.666666667 Combine like terms: 3.282952601 + -1.666666667 = 1.616285934 r = 1.616285934 Simplifying r = 1.616285934

Subproblem 2

r + 1.666666667 = -3.282952601 Simplifying r + 1.666666667 = -3.282952601 Reorder the terms: 1.666666667 + r = -3.282952601 Solving 1.666666667 + r = -3.282952601 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.666666667' to each side of the equation. 1.666666667 + -1.666666667 + r = -3.282952601 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + r = -3.282952601 + -1.666666667 r = -3.282952601 + -1.666666667 Combine like terms: -3.282952601 + -1.666666667 = -4.949619268 r = -4.949619268 Simplifying r = -4.949619268

Solution

The solution to the problem is based on the solutions from the subproblems. r = {1.616285934, -4.949619268}

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