r^2+14r-102=-9

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Solution for r^2+14r-102=-9 equation:


Simplifying
r2 + 14r + -102 = -9

Reorder the terms:
-102 + 14r + r2 = -9

Solving
-102 + 14r + r2 = -9

Solving for variable 'r'.

Reorder the terms:
-102 + 9 + 14r + r2 = -9 + 9

Combine like terms: -102 + 9 = -93
-93 + 14r + r2 = -9 + 9

Combine like terms: -9 + 9 = 0
-93 + 14r + r2 = 0

Begin completing the square.

Move the constant term to the right:

Add '93' to each side of the equation.
-93 + 14r + 93 + r2 = 0 + 93

Reorder the terms:
-93 + 93 + 14r + r2 = 0 + 93

Combine like terms: -93 + 93 = 0
0 + 14r + r2 = 0 + 93
14r + r2 = 0 + 93

Combine like terms: 0 + 93 = 93
14r + r2 = 93

The r term is 14r.  Take half its coefficient (7).
Square it (49) and add it to both sides.

Add '49' to each side of the equation.
14r + 49 + r2 = 93 + 49

Reorder the terms:
49 + 14r + r2 = 93 + 49

Combine like terms: 93 + 49 = 142
49 + 14r + r2 = 142

Factor a perfect square on the left side:
(r + 7)(r + 7) = 142

Calculate the square root of the right side: 11.916375288

Break this problem into two subproblems by setting 
(r + 7) equal to 11.916375288 and -11.916375288.

Subproblem 1

r + 7 = 11.916375288 Simplifying r + 7 = 11.916375288 Reorder the terms: 7 + r = 11.916375288 Solving 7 + r = 11.916375288 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-7' to each side of the equation. 7 + -7 + r = 11.916375288 + -7 Combine like terms: 7 + -7 = 0 0 + r = 11.916375288 + -7 r = 11.916375288 + -7 Combine like terms: 11.916375288 + -7 = 4.916375288 r = 4.916375288 Simplifying r = 4.916375288

Subproblem 2

r + 7 = -11.916375288 Simplifying r + 7 = -11.916375288 Reorder the terms: 7 + r = -11.916375288 Solving 7 + r = -11.916375288 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-7' to each side of the equation. 7 + -7 + r = -11.916375288 + -7 Combine like terms: 7 + -7 = 0 0 + r = -11.916375288 + -7 r = -11.916375288 + -7 Combine like terms: -11.916375288 + -7 = -18.916375288 r = -18.916375288 Simplifying r = -18.916375288

Solution

The solution to the problem is based on the solutions from the subproblems. r = {4.916375288, -18.916375288}

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