6k(k+4)+3=5(k^2-12)+8

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Solution for 6k(k+4)+3=5(k^2-12)+8 equation:


Simplifying
6k(k + 4) + 3 = 5(k2 + -12) + 8

Reorder the terms:
6k(4 + k) + 3 = 5(k2 + -12) + 8
(4 * 6k + k * 6k) + 3 = 5(k2 + -12) + 8
(24k + 6k2) + 3 = 5(k2 + -12) + 8

Reorder the terms:
3 + 24k + 6k2 = 5(k2 + -12) + 8

Reorder the terms:
3 + 24k + 6k2 = 5(-12 + k2) + 8
3 + 24k + 6k2 = (-12 * 5 + k2 * 5) + 8
3 + 24k + 6k2 = (-60 + 5k2) + 8

Reorder the terms:
3 + 24k + 6k2 = -60 + 8 + 5k2

Combine like terms: -60 + 8 = -52
3 + 24k + 6k2 = -52 + 5k2

Solving
3 + 24k + 6k2 = -52 + 5k2

Solving for variable 'k'.

Reorder the terms:
3 + 52 + 24k + 6k2 + -5k2 = -52 + 5k2 + 52 + -5k2

Combine like terms: 3 + 52 = 55
55 + 24k + 6k2 + -5k2 = -52 + 5k2 + 52 + -5k2

Combine like terms: 6k2 + -5k2 = 1k2
55 + 24k + 1k2 = -52 + 5k2 + 52 + -5k2

Reorder the terms:
55 + 24k + 1k2 = -52 + 52 + 5k2 + -5k2

Combine like terms: -52 + 52 = 0
55 + 24k + 1k2 = 0 + 5k2 + -5k2
55 + 24k + 1k2 = 5k2 + -5k2

Combine like terms: 5k2 + -5k2 = 0
55 + 24k + 1k2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-55' to each side of the equation.
55 + 24k + -55 + k2 = 0 + -55

Reorder the terms:
55 + -55 + 24k + k2 = 0 + -55

Combine like terms: 55 + -55 = 0
0 + 24k + k2 = 0 + -55
24k + k2 = 0 + -55

Combine like terms: 0 + -55 = -55
24k + k2 = -55

The k term is 24k.  Take half its coefficient (12).
Square it (144) and add it to both sides.

Add '144' to each side of the equation.
24k + 144 + k2 = -55 + 144

Reorder the terms:
144 + 24k + k2 = -55 + 144

Combine like terms: -55 + 144 = 89
144 + 24k + k2 = 89

Factor a perfect square on the left side:
(k + 12)(k + 12) = 89

Calculate the square root of the right side: 9.433981132

Break this problem into two subproblems by setting 
(k + 12) equal to 9.433981132 and -9.433981132.

Subproblem 1

k + 12 = 9.433981132 Simplifying k + 12 = 9.433981132 Reorder the terms: 12 + k = 9.433981132 Solving 12 + k = 9.433981132 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-12' to each side of the equation. 12 + -12 + k = 9.433981132 + -12 Combine like terms: 12 + -12 = 0 0 + k = 9.433981132 + -12 k = 9.433981132 + -12 Combine like terms: 9.433981132 + -12 = -2.566018868 k = -2.566018868 Simplifying k = -2.566018868

Subproblem 2

k + 12 = -9.433981132 Simplifying k + 12 = -9.433981132 Reorder the terms: 12 + k = -9.433981132 Solving 12 + k = -9.433981132 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-12' to each side of the equation. 12 + -12 + k = -9.433981132 + -12 Combine like terms: 12 + -12 = 0 0 + k = -9.433981132 + -12 k = -9.433981132 + -12 Combine like terms: -9.433981132 + -12 = -21.433981132 k = -21.433981132 Simplifying k = -21.433981132

Solution

The solution to the problem is based on the solutions from the subproblems. k = {-2.566018868, -21.433981132}

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