(7x-3)(3x-7)=180

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Solution for (7x-3)(3x-7)=180 equation:


Simplifying
(7x + -3)(3x + -7) = 180

Reorder the terms:
(-3 + 7x)(3x + -7) = 180

Reorder the terms:
(-3 + 7x)(-7 + 3x) = 180

Multiply (-3 + 7x) * (-7 + 3x)
(-3(-7 + 3x) + 7x * (-7 + 3x)) = 180
((-7 * -3 + 3x * -3) + 7x * (-7 + 3x)) = 180
((21 + -9x) + 7x * (-7 + 3x)) = 180
(21 + -9x + (-7 * 7x + 3x * 7x)) = 180
(21 + -9x + (-49x + 21x2)) = 180

Combine like terms: -9x + -49x = -58x
(21 + -58x + 21x2) = 180

Solving
21 + -58x + 21x2 = 180

Solving for variable 'x'.

Reorder the terms:
21 + -180 + -58x + 21x2 = 180 + -180

Combine like terms: 21 + -180 = -159
-159 + -58x + 21x2 = 180 + -180

Combine like terms: 180 + -180 = 0
-159 + -58x + 21x2 = 0

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

Divide each side by '21'.
-7.571428571 + -2.761904762x + x2 = 0

Move the constant term to the right:

Add '7.571428571' to each side of the equation.
-7.571428571 + -2.761904762x + 7.571428571 + x2 = 0 + 7.571428571

Reorder the terms:
-7.571428571 + 7.571428571 + -2.761904762x + x2 = 0 + 7.571428571

Combine like terms: -7.571428571 + 7.571428571 = 0.000000000
0.000000000 + -2.761904762x + x2 = 0 + 7.571428571
-2.761904762x + x2 = 0 + 7.571428571

Combine like terms: 0 + 7.571428571 = 7.571428571
-2.761904762x + x2 = 7.571428571

The x term is -2.761904762x.  Take half its coefficient (-1.380952381).
Square it (1.907029479) and add it to both sides.

Add '1.907029479' to each side of the equation.
-2.761904762x + 1.907029479 + x2 = 7.571428571 + 1.907029479

Reorder the terms:
1.907029479 + -2.761904762x + x2 = 7.571428571 + 1.907029479

Combine like terms: 7.571428571 + 1.907029479 = 9.47845805
1.907029479 + -2.761904762x + x2 = 9.47845805

Factor a perfect square on the left side:
(x + -1.380952381)(x + -1.380952381) = 9.47845805

Calculate the square root of the right side: 3.078710452

Break this problem into two subproblems by setting 
(x + -1.380952381) equal to 3.078710452 and -3.078710452.

Subproblem 1

x + -1.380952381 = 3.078710452 Simplifying x + -1.380952381 = 3.078710452 Reorder the terms: -1.380952381 + x = 3.078710452 Solving -1.380952381 + x = 3.078710452 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.380952381' to each side of the equation. -1.380952381 + 1.380952381 + x = 3.078710452 + 1.380952381 Combine like terms: -1.380952381 + 1.380952381 = 0.000000000 0.000000000 + x = 3.078710452 + 1.380952381 x = 3.078710452 + 1.380952381 Combine like terms: 3.078710452 + 1.380952381 = 4.459662833 x = 4.459662833 Simplifying x = 4.459662833

Subproblem 2

x + -1.380952381 = -3.078710452 Simplifying x + -1.380952381 = -3.078710452 Reorder the terms: -1.380952381 + x = -3.078710452 Solving -1.380952381 + x = -3.078710452 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.380952381' to each side of the equation. -1.380952381 + 1.380952381 + x = -3.078710452 + 1.380952381 Combine like terms: -1.380952381 + 1.380952381 = 0.000000000 0.000000000 + x = -3.078710452 + 1.380952381 x = -3.078710452 + 1.380952381 Combine like terms: -3.078710452 + 1.380952381 = -1.697758071 x = -1.697758071 Simplifying x = -1.697758071

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.459662833, -1.697758071}

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