6r^2-10r=23

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Solution for 6r^2-10r=23 equation:


Simplifying
6r2 + -10r = 23

Reorder the terms:
-10r + 6r2 = 23

Solving
-10r + 6r2 = 23

Solving for variable 'r'.

Reorder the terms:
-23 + -10r + 6r2 = 23 + -23

Combine like terms: 23 + -23 = 0
-23 + -10r + 6r2 = 0

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

Divide each side by '6'.
-3.833333333 + -1.666666667r + r2 = 0

Move the constant term to the right:

Add '3.833333333' to each side of the equation.
-3.833333333 + -1.666666667r + 3.833333333 + r2 = 0 + 3.833333333

Reorder the terms:
-3.833333333 + 3.833333333 + -1.666666667r + r2 = 0 + 3.833333333

Combine like terms: -3.833333333 + 3.833333333 = 0.000000000
0.000000000 + -1.666666667r + r2 = 0 + 3.833333333
-1.666666667r + r2 = 0 + 3.833333333

Combine like terms: 0 + 3.833333333 = 3.833333333
-1.666666667r + r2 = 3.833333333

The r term is -1.666666667r.  Take half its coefficient (-0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
-1.666666667r + 0.6944444447 + r2 = 3.833333333 + 0.6944444447

Reorder the terms:
0.6944444447 + -1.666666667r + r2 = 3.833333333 + 0.6944444447

Combine like terms: 3.833333333 + 0.6944444447 = 4.5277777777
0.6944444447 + -1.666666667r + r2 = 4.5277777777

Factor a perfect square on the left side:
(r + -0.8333333335)(r + -0.8333333335) = 4.5277777777

Calculate the square root of the right side: 2.127857556

Break this problem into two subproblems by setting 
(r + -0.8333333335) equal to 2.127857556 and -2.127857556.

Subproblem 1

r + -0.8333333335 = 2.127857556 Simplifying r + -0.8333333335 = 2.127857556 Reorder the terms: -0.8333333335 + r = 2.127857556 Solving -0.8333333335 + r = 2.127857556 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + r = 2.127857556 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + r = 2.127857556 + 0.8333333335 r = 2.127857556 + 0.8333333335 Combine like terms: 2.127857556 + 0.8333333335 = 2.9611908895 r = 2.9611908895 Simplifying r = 2.9611908895

Subproblem 2

r + -0.8333333335 = -2.127857556 Simplifying r + -0.8333333335 = -2.127857556 Reorder the terms: -0.8333333335 + r = -2.127857556 Solving -0.8333333335 + r = -2.127857556 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + r = -2.127857556 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + r = -2.127857556 + 0.8333333335 r = -2.127857556 + 0.8333333335 Combine like terms: -2.127857556 + 0.8333333335 = -1.2945242225 r = -1.2945242225 Simplifying r = -1.2945242225

Solution

The solution to the problem is based on the solutions from the subproblems. r = {2.9611908895, -1.2945242225}

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