7x^2-16x-112=0

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Solution for 7x^2-16x-112=0 equation:


Simplifying
7x2 + -16x + -112 = 0

Reorder the terms:
-112 + -16x + 7x2 = 0

Solving
-112 + -16x + 7x2 = 0

Solving for variable 'x'.

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

Divide each side by '7'.
-16 + -2.285714286x + x2 = 0

Move the constant term to the right:

Add '16' to each side of the equation.
-16 + -2.285714286x + 16 + x2 = 0 + 16

Reorder the terms:
-16 + 16 + -2.285714286x + x2 = 0 + 16

Combine like terms: -16 + 16 = 0
0 + -2.285714286x + x2 = 0 + 16
-2.285714286x + x2 = 0 + 16

Combine like terms: 0 + 16 = 16
-2.285714286x + x2 = 16

The x term is -2.285714286x.  Take half its coefficient (-1.142857143).
Square it (1.306122449) and add it to both sides.

Add '1.306122449' to each side of the equation.
-2.285714286x + 1.306122449 + x2 = 16 + 1.306122449

Reorder the terms:
1.306122449 + -2.285714286x + x2 = 16 + 1.306122449

Combine like terms: 16 + 1.306122449 = 17.306122449
1.306122449 + -2.285714286x + x2 = 17.306122449

Factor a perfect square on the left side:
(x + -1.142857143)(x + -1.142857143) = 17.306122449

Calculate the square root of the right side: 4.160062794

Break this problem into two subproblems by setting 
(x + -1.142857143) equal to 4.160062794 and -4.160062794.

Subproblem 1

x + -1.142857143 = 4.160062794 Simplifying x + -1.142857143 = 4.160062794 Reorder the terms: -1.142857143 + x = 4.160062794 Solving -1.142857143 + x = 4.160062794 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.142857143' to each side of the equation. -1.142857143 + 1.142857143 + x = 4.160062794 + 1.142857143 Combine like terms: -1.142857143 + 1.142857143 = 0.000000000 0.000000000 + x = 4.160062794 + 1.142857143 x = 4.160062794 + 1.142857143 Combine like terms: 4.160062794 + 1.142857143 = 5.302919937 x = 5.302919937 Simplifying x = 5.302919937

Subproblem 2

x + -1.142857143 = -4.160062794 Simplifying x + -1.142857143 = -4.160062794 Reorder the terms: -1.142857143 + x = -4.160062794 Solving -1.142857143 + x = -4.160062794 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.142857143' to each side of the equation. -1.142857143 + 1.142857143 + x = -4.160062794 + 1.142857143 Combine like terms: -1.142857143 + 1.142857143 = 0.000000000 0.000000000 + x = -4.160062794 + 1.142857143 x = -4.160062794 + 1.142857143 Combine like terms: -4.160062794 + 1.142857143 = -3.017205651 x = -3.017205651 Simplifying x = -3.017205651

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.302919937, -3.017205651}

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