r^2+3r-7=0

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Solution for r^2+3r-7=0 equation:


Simplifying
r2 + 3r + -7 = 0

Reorder the terms:
-7 + 3r + r2 = 0

Solving
-7 + 3r + r2 = 0

Solving for variable 'r'.

Begin completing the square.

Move the constant term to the right:

Add '7' to each side of the equation.
-7 + 3r + 7 + r2 = 0 + 7

Reorder the terms:
-7 + 7 + 3r + r2 = 0 + 7

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

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

The r term is 3r.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3r + 2.25 + r2 = 7 + 2.25

Reorder the terms:
2.25 + 3r + r2 = 7 + 2.25

Combine like terms: 7 + 2.25 = 9.25
2.25 + 3r + r2 = 9.25

Factor a perfect square on the left side:
(r + 1.5)(r + 1.5) = 9.25

Calculate the square root of the right side: 3.041381265

Break this problem into two subproblems by setting 
(r + 1.5) equal to 3.041381265 and -3.041381265.

Subproblem 1

r + 1.5 = 3.041381265 Simplifying r + 1.5 = 3.041381265 Reorder the terms: 1.5 + r = 3.041381265 Solving 1.5 + r = 3.041381265 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + r = 3.041381265 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + r = 3.041381265 + -1.5 r = 3.041381265 + -1.5 Combine like terms: 3.041381265 + -1.5 = 1.541381265 r = 1.541381265 Simplifying r = 1.541381265

Subproblem 2

r + 1.5 = -3.041381265 Simplifying r + 1.5 = -3.041381265 Reorder the terms: 1.5 + r = -3.041381265 Solving 1.5 + r = -3.041381265 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + r = -3.041381265 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + r = -3.041381265 + -1.5 r = -3.041381265 + -1.5 Combine like terms: -3.041381265 + -1.5 = -4.541381265 r = -4.541381265 Simplifying r = -4.541381265

Solution

The solution to the problem is based on the solutions from the subproblems. r = {1.541381265, -4.541381265}

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