7n^2-6n-5=(3n+1)(4n+1)

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Solution for 7n^2-6n-5=(3n+1)(4n+1) equation:


Simplifying
7n2 + -6n + -5 = (3n + 1)(4n + 1)

Reorder the terms:
-5 + -6n + 7n2 = (3n + 1)(4n + 1)

Reorder the terms:
-5 + -6n + 7n2 = (1 + 3n)(4n + 1)

Reorder the terms:
-5 + -6n + 7n2 = (1 + 3n)(1 + 4n)

Multiply (1 + 3n) * (1 + 4n)
-5 + -6n + 7n2 = (1(1 + 4n) + 3n * (1 + 4n))
-5 + -6n + 7n2 = ((1 * 1 + 4n * 1) + 3n * (1 + 4n))
-5 + -6n + 7n2 = ((1 + 4n) + 3n * (1 + 4n))
-5 + -6n + 7n2 = (1 + 4n + (1 * 3n + 4n * 3n))
-5 + -6n + 7n2 = (1 + 4n + (3n + 12n2))

Combine like terms: 4n + 3n = 7n
-5 + -6n + 7n2 = (1 + 7n + 12n2)

Solving
-5 + -6n + 7n2 = 1 + 7n + 12n2

Solving for variable 'n'.

Reorder the terms:
-5 + -1 + -6n + -7n + 7n2 + -12n2 = 1 + 7n + 12n2 + -1 + -7n + -12n2

Combine like terms: -5 + -1 = -6
-6 + -6n + -7n + 7n2 + -12n2 = 1 + 7n + 12n2 + -1 + -7n + -12n2

Combine like terms: -6n + -7n = -13n
-6 + -13n + 7n2 + -12n2 = 1 + 7n + 12n2 + -1 + -7n + -12n2

Combine like terms: 7n2 + -12n2 = -5n2
-6 + -13n + -5n2 = 1 + 7n + 12n2 + -1 + -7n + -12n2

Reorder the terms:
-6 + -13n + -5n2 = 1 + -1 + 7n + -7n + 12n2 + -12n2

Combine like terms: 1 + -1 = 0
-6 + -13n + -5n2 = 0 + 7n + -7n + 12n2 + -12n2
-6 + -13n + -5n2 = 7n + -7n + 12n2 + -12n2

Combine like terms: 7n + -7n = 0
-6 + -13n + -5n2 = 0 + 12n2 + -12n2
-6 + -13n + -5n2 = 12n2 + -12n2

Combine like terms: 12n2 + -12n2 = 0
-6 + -13n + -5n2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(6 + 13n + 5n2) = 0

Factor a trinomial.
-1((2 + n)(3 + 5n)) = 0

Ignore the factor -1.

Subproblem 1

Set the factor '(2 + n)' equal to zero and attempt to solve: Simplifying 2 + n = 0 Solving 2 + n = 0 Move all terms containing n to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + n = 0 + -2 Combine like terms: 2 + -2 = 0 0 + n = 0 + -2 n = 0 + -2 Combine like terms: 0 + -2 = -2 n = -2 Simplifying n = -2

Subproblem 2

Set the factor '(3 + 5n)' equal to zero and attempt to solve: Simplifying 3 + 5n = 0 Solving 3 + 5n = 0 Move all terms containing n to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + 5n = 0 + -3 Combine like terms: 3 + -3 = 0 0 + 5n = 0 + -3 5n = 0 + -3 Combine like terms: 0 + -3 = -3 5n = -3 Divide each side by '5'. n = -0.6 Simplifying n = -0.6

Solution

n = {-2, -0.6}

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