4n(n+6)=22

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Solution for 4n(n+6)=22 equation:


Simplifying
4n(n + 6) = 22

Reorder the terms:
4n(6 + n) = 22
(6 * 4n + n * 4n) = 22
(24n + 4n2) = 22

Solving
24n + 4n2 = 22

Solving for variable 'n'.

Reorder the terms:
-22 + 24n + 4n2 = 22 + -22

Combine like terms: 22 + -22 = 0
-22 + 24n + 4n2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-11 + 12n + 2n2) = 0

Ignore the factor 2.

Subproblem 1

Set the factor '(-11 + 12n + 2n2)' equal to zero and attempt to solve: Simplifying -11 + 12n + 2n2 = 0 Solving -11 + 12n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -5.5 + 6n + n2 = 0 Move the constant term to the right: Add '5.5' to each side of the equation. -5.5 + 6n + 5.5 + n2 = 0 + 5.5 Reorder the terms: -5.5 + 5.5 + 6n + n2 = 0 + 5.5 Combine like terms: -5.5 + 5.5 = 0.0 0.0 + 6n + n2 = 0 + 5.5 6n + n2 = 0 + 5.5 Combine like terms: 0 + 5.5 = 5.5 6n + n2 = 5.5 The n term is 6n. Take half its coefficient (3). Square it (9) and add it to both sides. Add '9' to each side of the equation. 6n + 9 + n2 = 5.5 + 9 Reorder the terms: 9 + 6n + n2 = 5.5 + 9 Combine like terms: 5.5 + 9 = 14.5 9 + 6n + n2 = 14.5 Factor a perfect square on the left side: (n + 3)(n + 3) = 14.5 Calculate the square root of the right side: 3.807886553 Break this problem into two subproblems by setting (n + 3) equal to 3.807886553 and -3.807886553.

Subproblem 1

n + 3 = 3.807886553 Simplifying n + 3 = 3.807886553 Reorder the terms: 3 + n = 3.807886553 Solving 3 + n = 3.807886553 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + n = 3.807886553 + -3 Combine like terms: 3 + -3 = 0 0 + n = 3.807886553 + -3 n = 3.807886553 + -3 Combine like terms: 3.807886553 + -3 = 0.807886553 n = 0.807886553 Simplifying n = 0.807886553

Subproblem 2

n + 3 = -3.807886553 Simplifying n + 3 = -3.807886553 Reorder the terms: 3 + n = -3.807886553 Solving 3 + n = -3.807886553 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + n = -3.807886553 + -3 Combine like terms: 3 + -3 = 0 0 + n = -3.807886553 + -3 n = -3.807886553 + -3 Combine like terms: -3.807886553 + -3 = -6.807886553 n = -6.807886553 Simplifying n = -6.807886553

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.807886553, -6.807886553}

Solution

n = {0.807886553, -6.807886553}

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