(x-7)(7x+5)=78

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Solution for (x-7)(7x+5)=78 equation:


Simplifying
(x + -7)(7x + 5) = 78

Reorder the terms:
(-7 + x)(7x + 5) = 78

Reorder the terms:
(-7 + x)(5 + 7x) = 78

Multiply (-7 + x) * (5 + 7x)
(-7(5 + 7x) + x(5 + 7x)) = 78
((5 * -7 + 7x * -7) + x(5 + 7x)) = 78
((-35 + -49x) + x(5 + 7x)) = 78
(-35 + -49x + (5 * x + 7x * x)) = 78
(-35 + -49x + (5x + 7x2)) = 78

Combine like terms: -49x + 5x = -44x
(-35 + -44x + 7x2) = 78

Solving
-35 + -44x + 7x2 = 78

Solving for variable 'x'.

Reorder the terms:
-35 + -78 + -44x + 7x2 = 78 + -78

Combine like terms: -35 + -78 = -113
-113 + -44x + 7x2 = 78 + -78

Combine like terms: 78 + -78 = 0
-113 + -44x + 7x2 = 0

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

Divide each side by '7'.
-16.14285714 + -6.285714286x + x2 = 0

Move the constant term to the right:

Add '16.14285714' to each side of the equation.
-16.14285714 + -6.285714286x + 16.14285714 + x2 = 0 + 16.14285714

Reorder the terms:
-16.14285714 + 16.14285714 + -6.285714286x + x2 = 0 + 16.14285714

Combine like terms: -16.14285714 + 16.14285714 = 0.00000000
0.00000000 + -6.285714286x + x2 = 0 + 16.14285714
-6.285714286x + x2 = 0 + 16.14285714

Combine like terms: 0 + 16.14285714 = 16.14285714
-6.285714286x + x2 = 16.14285714

The x term is -6.285714286x.  Take half its coefficient (-3.142857143).
Square it (9.877551021) and add it to both sides.

Add '9.877551021' to each side of the equation.
-6.285714286x + 9.877551021 + x2 = 16.14285714 + 9.877551021

Reorder the terms:
9.877551021 + -6.285714286x + x2 = 16.14285714 + 9.877551021

Combine like terms: 16.14285714 + 9.877551021 = 26.020408161
9.877551021 + -6.285714286x + x2 = 26.020408161

Factor a perfect square on the left side:
(x + -3.142857143)(x + -3.142857143) = 26.020408161

Calculate the square root of the right side: 5.101020306

Break this problem into two subproblems by setting 
(x + -3.142857143) equal to 5.101020306 and -5.101020306.

Subproblem 1

x + -3.142857143 = 5.101020306 Simplifying x + -3.142857143 = 5.101020306 Reorder the terms: -3.142857143 + x = 5.101020306 Solving -3.142857143 + x = 5.101020306 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.142857143' to each side of the equation. -3.142857143 + 3.142857143 + x = 5.101020306 + 3.142857143 Combine like terms: -3.142857143 + 3.142857143 = 0.000000000 0.000000000 + x = 5.101020306 + 3.142857143 x = 5.101020306 + 3.142857143 Combine like terms: 5.101020306 + 3.142857143 = 8.243877449 x = 8.243877449 Simplifying x = 8.243877449

Subproblem 2

x + -3.142857143 = -5.101020306 Simplifying x + -3.142857143 = -5.101020306 Reorder the terms: -3.142857143 + x = -5.101020306 Solving -3.142857143 + x = -5.101020306 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.142857143' to each side of the equation. -3.142857143 + 3.142857143 + x = -5.101020306 + 3.142857143 Combine like terms: -3.142857143 + 3.142857143 = 0.000000000 0.000000000 + x = -5.101020306 + 3.142857143 x = -5.101020306 + 3.142857143 Combine like terms: -5.101020306 + 3.142857143 = -1.958163163 x = -1.958163163 Simplifying x = -1.958163163

Solution

The solution to the problem is based on the solutions from the subproblems. x = {8.243877449, -1.958163163}

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