simplify (n^2-n-12)/(n^2-4n) — step-by-step solution

The original denominators must be nonzero:

n \in ℝ \ {0, 4}

1) Let's simplify:

\frac{n^{2} - n - 12}{n^{2} - 4 n}

2) Let's factor in

ℝ

n^{2} - n - 12

Given the quadratic form

a n^{2} + b n + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 1

b = -1

and

c = -12

Δ = {(-1)}^{2} - 4 \cdot 1 \cdot (-12)

ⓘ Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product:

{(-1)}^{2}

(-1) \cdot (-1)

1

Δ = 1 - 4 \cdot (-12)

ⓘ Multiplications are done before additions and subtractions. Two minus signs cancel out, the product of two negative numbers is positive:

- 4 \cdot (-12)

4 \cdot 12

4 \cdot (10 + 2)

4 \cdot 10 + 4 \cdot 2

40 + 8

48

Δ = 1 + 48

ⓘ Start from

1

and move forward

48

on the number line:

1 + 48

49

Δ = 49

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

n_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-1) - \sqrt{49}}{2 \cdot 1}

n_{1} = \frac{1 - \sqrt{49}}{2}

ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{49}

7

;

7^{2}

49

n_{1} = \frac{1 - 7}{2}

ⓘ The numerator and the denominator are computed before the division:

1 - 7

-6

n_{1} = - \frac{6}{2}

ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

- \frac{6}{2}

-3

n_{1} = -3

The quadratic formula gives

n_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-1) + \sqrt{49}}{2 \cdot 1}

n_{2} = \frac{1 + \sqrt{49}}{2}

ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{49}

7

;

7^{2}

49

n_{2} = \frac{1 + 7}{2}

ⓘ The numerator and the denominator are computed before the division:

1 + 7

8

n_{2} = \frac{8}{2}

ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

\frac{8}{2}

4

n_{2} = 4

Factors identified:

(n + 3) \cdot (n - 4)

3) Let's factorize in

ℝ

n^{2} - 4 n

You can factorize this expression into:

n \cdot (n - 4)

ⓘ The original denominator must remain nonzero. The fraction is simplified by dividing the numerator and the denominator by their common factor:

\frac{n^{2} - n - 12}{n^{2} - 4 n}

\frac{n + 3}{n}

\frac{n + 3}{n}

ⅰ

\frac{n + 3}{n}

1 + \frac{3}{n}

1 + \frac{3}{n}

▶Simplified form:

\frac{3}{n} + 1