(4r-12)(7r-16)=-1

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Solution for (4r-12)(7r-16)=-1 equation:


Simplifying
(4r + -12)(7r + -16) = -1

Reorder the terms:
(-12 + 4r)(7r + -16) = -1

Reorder the terms:
(-12 + 4r)(-16 + 7r) = -1

Multiply (-12 + 4r) * (-16 + 7r)
(-12(-16 + 7r) + 4r * (-16 + 7r)) = -1
((-16 * -12 + 7r * -12) + 4r * (-16 + 7r)) = -1
((192 + -84r) + 4r * (-16 + 7r)) = -1
(192 + -84r + (-16 * 4r + 7r * 4r)) = -1
(192 + -84r + (-64r + 28r2)) = -1

Combine like terms: -84r + -64r = -148r
(192 + -148r + 28r2) = -1

Solving
192 + -148r + 28r2 = -1

Solving for variable 'r'.

Reorder the terms:
192 + 1 + -148r + 28r2 = -1 + 1

Combine like terms: 192 + 1 = 193
193 + -148r + 28r2 = -1 + 1

Combine like terms: -1 + 1 = 0
193 + -148r + 28r2 = 0

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

Divide each side by '28'.
6.892857143 + -5.285714286r + r2 = 0

Move the constant term to the right:

Add '-6.892857143' to each side of the equation.
6.892857143 + -5.285714286r + -6.892857143 + r2 = 0 + -6.892857143

Reorder the terms:
6.892857143 + -6.892857143 + -5.285714286r + r2 = 0 + -6.892857143

Combine like terms: 6.892857143 + -6.892857143 = 0.000000000
0.000000000 + -5.285714286r + r2 = 0 + -6.892857143
-5.285714286r + r2 = 0 + -6.892857143

Combine like terms: 0 + -6.892857143 = -6.892857143
-5.285714286r + r2 = -6.892857143

The r term is -5.285714286r.  Take half its coefficient (-2.642857143).
Square it (6.984693878) and add it to both sides.

Add '6.984693878' to each side of the equation.
-5.285714286r + 6.984693878 + r2 = -6.892857143 + 6.984693878

Reorder the terms:
6.984693878 + -5.285714286r + r2 = -6.892857143 + 6.984693878

Combine like terms: -6.892857143 + 6.984693878 = 0.091836735
6.984693878 + -5.285714286r + r2 = 0.091836735

Factor a perfect square on the left side:
(r + -2.642857143)(r + -2.642857143) = 0.091836735

Calculate the square root of the right side: 0.303045764

Break this problem into two subproblems by setting 
(r + -2.642857143) equal to 0.303045764 and -0.303045764.

Subproblem 1

r + -2.642857143 = 0.303045764 Simplifying r + -2.642857143 = 0.303045764 Reorder the terms: -2.642857143 + r = 0.303045764 Solving -2.642857143 + r = 0.303045764 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '2.642857143' to each side of the equation. -2.642857143 + 2.642857143 + r = 0.303045764 + 2.642857143 Combine like terms: -2.642857143 + 2.642857143 = 0.000000000 0.000000000 + r = 0.303045764 + 2.642857143 r = 0.303045764 + 2.642857143 Combine like terms: 0.303045764 + 2.642857143 = 2.945902907 r = 2.945902907 Simplifying r = 2.945902907

Subproblem 2

r + -2.642857143 = -0.303045764 Simplifying r + -2.642857143 = -0.303045764 Reorder the terms: -2.642857143 + r = -0.303045764 Solving -2.642857143 + r = -0.303045764 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '2.642857143' to each side of the equation. -2.642857143 + 2.642857143 + r = -0.303045764 + 2.642857143 Combine like terms: -2.642857143 + 2.642857143 = 0.000000000 0.000000000 + r = -0.303045764 + 2.642857143 r = -0.303045764 + 2.642857143 Combine like terms: -0.303045764 + 2.642857143 = 2.339811379 r = 2.339811379 Simplifying r = 2.339811379

Solution

The solution to the problem is based on the solutions from the subproblems. r = {2.945902907, 2.339811379}

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