r^2+4r+7=14

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Solution for r^2+4r+7=14 equation:


Simplifying
r2 + 4r + 7 = 14

Reorder the terms:
7 + 4r + r2 = 14

Solving
7 + 4r + r2 = 14

Solving for variable 'r'.

Reorder the terms:
7 + -14 + 4r + r2 = 14 + -14

Combine like terms: 7 + -14 = -7
-7 + 4r + r2 = 14 + -14

Combine like terms: 14 + -14 = 0
-7 + 4r + r2 = 0

Begin completing the square.

Move the constant term to the right:

Add '7' to each side of the equation.
-7 + 4r + 7 + r2 = 0 + 7

Reorder the terms:
-7 + 7 + 4r + r2 = 0 + 7

Combine like terms: -7 + 7 = 0
0 + 4r + r2 = 0 + 7
4r + r2 = 0 + 7

Combine like terms: 0 + 7 = 7
4r + r2 = 7

The r term is 4r.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4r + 4 + r2 = 7 + 4

Reorder the terms:
4 + 4r + r2 = 7 + 4

Combine like terms: 7 + 4 = 11
4 + 4r + r2 = 11

Factor a perfect square on the left side:
(r + 2)(r + 2) = 11

Calculate the square root of the right side: 3.31662479

Break this problem into two subproblems by setting 
(r + 2) equal to 3.31662479 and -3.31662479.

Subproblem 1

r + 2 = 3.31662479 Simplifying r + 2 = 3.31662479 Reorder the terms: 2 + r = 3.31662479 Solving 2 + r = 3.31662479 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + r = 3.31662479 + -2 Combine like terms: 2 + -2 = 0 0 + r = 3.31662479 + -2 r = 3.31662479 + -2 Combine like terms: 3.31662479 + -2 = 1.31662479 r = 1.31662479 Simplifying r = 1.31662479

Subproblem 2

r + 2 = -3.31662479 Simplifying r + 2 = -3.31662479 Reorder the terms: 2 + r = -3.31662479 Solving 2 + r = -3.31662479 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + r = -3.31662479 + -2 Combine like terms: 2 + -2 = 0 0 + r = -3.31662479 + -2 r = -3.31662479 + -2 Combine like terms: -3.31662479 + -2 = -5.31662479 r = -5.31662479 Simplifying r = -5.31662479

Solution

The solution to the problem is based on the solutions from the subproblems. r = {1.31662479, -5.31662479}

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