7x^2=8x+16

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Solution for 7x^2=8x+16 equation:


Simplifying
7x2 = 8x + 16

Reorder the terms:
7x2 = 16 + 8x

Solving
7x2 = 16 + 8x

Solving for variable 'x'.

Reorder the terms:
-16 + -8x + 7x2 = 16 + 8x + -16 + -8x

Reorder the terms:
-16 + -8x + 7x2 = 16 + -16 + 8x + -8x

Combine like terms: 16 + -16 = 0
-16 + -8x + 7x2 = 0 + 8x + -8x
-16 + -8x + 7x2 = 8x + -8x

Combine like terms: 8x + -8x = 0
-16 + -8x + 7x2 = 0

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

Divide each side by '7'.
-2.285714286 + -1.142857143x + x2 = 0

Move the constant term to the right:

Add '2.285714286' to each side of the equation.
-2.285714286 + -1.142857143x + 2.285714286 + x2 = 0 + 2.285714286

Reorder the terms:
-2.285714286 + 2.285714286 + -1.142857143x + x2 = 0 + 2.285714286

Combine like terms: -2.285714286 + 2.285714286 = 0.000000000
0.000000000 + -1.142857143x + x2 = 0 + 2.285714286
-1.142857143x + x2 = 0 + 2.285714286

Combine like terms: 0 + 2.285714286 = 2.285714286
-1.142857143x + x2 = 2.285714286

The x term is -1.142857143x.  Take half its coefficient (-0.5714285715).
Square it (0.3265306123) and add it to both sides.

Add '0.3265306123' to each side of the equation.
-1.142857143x + 0.3265306123 + x2 = 2.285714286 + 0.3265306123

Reorder the terms:
0.3265306123 + -1.142857143x + x2 = 2.285714286 + 0.3265306123

Combine like terms: 2.285714286 + 0.3265306123 = 2.6122448983
0.3265306123 + -1.142857143x + x2 = 2.6122448983

Factor a perfect square on the left side:
(x + -0.5714285715)(x + -0.5714285715) = 2.6122448983

Calculate the square root of the right side: 1.616244071

Break this problem into two subproblems by setting 
(x + -0.5714285715) equal to 1.616244071 and -1.616244071.

Subproblem 1

x + -0.5714285715 = 1.616244071 Simplifying x + -0.5714285715 = 1.616244071 Reorder the terms: -0.5714285715 + x = 1.616244071 Solving -0.5714285715 + x = 1.616244071 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5714285715' to each side of the equation. -0.5714285715 + 0.5714285715 + x = 1.616244071 + 0.5714285715 Combine like terms: -0.5714285715 + 0.5714285715 = 0.0000000000 0.0000000000 + x = 1.616244071 + 0.5714285715 x = 1.616244071 + 0.5714285715 Combine like terms: 1.616244071 + 0.5714285715 = 2.1876726425 x = 2.1876726425 Simplifying x = 2.1876726425

Subproblem 2

x + -0.5714285715 = -1.616244071 Simplifying x + -0.5714285715 = -1.616244071 Reorder the terms: -0.5714285715 + x = -1.616244071 Solving -0.5714285715 + x = -1.616244071 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5714285715' to each side of the equation. -0.5714285715 + 0.5714285715 + x = -1.616244071 + 0.5714285715 Combine like terms: -0.5714285715 + 0.5714285715 = 0.0000000000 0.0000000000 + x = -1.616244071 + 0.5714285715 x = -1.616244071 + 0.5714285715 Combine like terms: -1.616244071 + 0.5714285715 = -1.0448154995 x = -1.0448154995 Simplifying x = -1.0448154995

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.1876726425, -1.0448154995}

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