k^2-8k-84=9

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Solution for k^2-8k-84=9 equation:


Simplifying
k2 + -8k + -84 = 9

Reorder the terms:
-84 + -8k + k2 = 9

Solving
-84 + -8k + k2 = 9

Solving for variable 'k'.

Reorder the terms:
-84 + -9 + -8k + k2 = 9 + -9

Combine like terms: -84 + -9 = -93
-93 + -8k + k2 = 9 + -9

Combine like terms: 9 + -9 = 0
-93 + -8k + k2 = 0

Begin completing the square.

Move the constant term to the right:

Add '93' to each side of the equation.
-93 + -8k + 93 + k2 = 0 + 93

Reorder the terms:
-93 + 93 + -8k + k2 = 0 + 93

Combine like terms: -93 + 93 = 0
0 + -8k + k2 = 0 + 93
-8k + k2 = 0 + 93

Combine like terms: 0 + 93 = 93
-8k + k2 = 93

The k term is -8k.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8k + 16 + k2 = 93 + 16

Reorder the terms:
16 + -8k + k2 = 93 + 16

Combine like terms: 93 + 16 = 109
16 + -8k + k2 = 109

Factor a perfect square on the left side:
(k + -4)(k + -4) = 109

Calculate the square root of the right side: 10.440306509

Break this problem into two subproblems by setting 
(k + -4) equal to 10.440306509 and -10.440306509.

Subproblem 1

k + -4 = 10.440306509 Simplifying k + -4 = 10.440306509 Reorder the terms: -4 + k = 10.440306509 Solving -4 + k = 10.440306509 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + k = 10.440306509 + 4 Combine like terms: -4 + 4 = 0 0 + k = 10.440306509 + 4 k = 10.440306509 + 4 Combine like terms: 10.440306509 + 4 = 14.440306509 k = 14.440306509 Simplifying k = 14.440306509

Subproblem 2

k + -4 = -10.440306509 Simplifying k + -4 = -10.440306509 Reorder the terms: -4 + k = -10.440306509 Solving -4 + k = -10.440306509 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + k = -10.440306509 + 4 Combine like terms: -4 + 4 = 0 0 + k = -10.440306509 + 4 k = -10.440306509 + 4 Combine like terms: -10.440306509 + 4 = -6.440306509 k = -6.440306509 Simplifying k = -6.440306509

Solution

The solution to the problem is based on the solutions from the subproblems. k = {14.440306509, -6.440306509}

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