5n+4n(n-3)=24

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Solution for 5n+4n(n-3)=24 equation:


Simplifying
5n + 4n(n + -3) = 24

Reorder the terms:
5n + 4n(-3 + n) = 24
5n + (-3 * 4n + n * 4n) = 24
5n + (-12n + 4n2) = 24

Combine like terms: 5n + -12n = -7n
-7n + 4n2 = 24

Solving
-7n + 4n2 = 24

Solving for variable 'n'.

Reorder the terms:
-24 + -7n + 4n2 = 24 + -24

Combine like terms: 24 + -24 = 0
-24 + -7n + 4n2 = 0

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

Divide each side by '4'.
-6 + -1.75n + n2 = 0

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + -1.75n + 6 + n2 = 0 + 6

Reorder the terms:
-6 + 6 + -1.75n + n2 = 0 + 6

Combine like terms: -6 + 6 = 0
0 + -1.75n + n2 = 0 + 6
-1.75n + n2 = 0 + 6

Combine like terms: 0 + 6 = 6
-1.75n + n2 = 6

The n term is -1.75n.  Take half its coefficient (-0.875).
Square it (0.765625) and add it to both sides.

Add '0.765625' to each side of the equation.
-1.75n + 0.765625 + n2 = 6 + 0.765625

Reorder the terms:
0.765625 + -1.75n + n2 = 6 + 0.765625

Combine like terms: 6 + 0.765625 = 6.765625
0.765625 + -1.75n + n2 = 6.765625

Factor a perfect square on the left side:
(n + -0.875)(n + -0.875) = 6.765625

Calculate the square root of the right side: 2.601081506

Break this problem into two subproblems by setting 
(n + -0.875) equal to 2.601081506 and -2.601081506.

Subproblem 1

n + -0.875 = 2.601081506 Simplifying n + -0.875 = 2.601081506 Reorder the terms: -0.875 + n = 2.601081506 Solving -0.875 + n = 2.601081506 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + n = 2.601081506 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + n = 2.601081506 + 0.875 n = 2.601081506 + 0.875 Combine like terms: 2.601081506 + 0.875 = 3.476081506 n = 3.476081506 Simplifying n = 3.476081506

Subproblem 2

n + -0.875 = -2.601081506 Simplifying n + -0.875 = -2.601081506 Reorder the terms: -0.875 + n = -2.601081506 Solving -0.875 + n = -2.601081506 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + n = -2.601081506 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + n = -2.601081506 + 0.875 n = -2.601081506 + 0.875 Combine like terms: -2.601081506 + 0.875 = -1.726081506 n = -1.726081506 Simplifying n = -1.726081506

Solution

The solution to the problem is based on the solutions from the subproblems. n = {3.476081506, -1.726081506}

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