1Cademy - Subtracting $$\frac{2n^2+8n-1}{n^2-1}

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Subtracting $\frac{m^2-6m}{m^2-1} - \frac{3m+2}{1-m^2}$

Example

Subtracting $\frac{2n^2+8n-1}{n^2-1} - \frac{n^2-7n-1}{1-n^2}$

Subtract two rational expressions whose denominators are opposites, then combine like terms and simplify:

$\frac{2n^2 + 8n - 1}{n^2 - 1} - \frac{n^2 - 7n - 1}{1 - n^2}$

Step 1 — Recognize the opposite denominators. The denominators $n^2 - 1$ and $1 - n^2$ are opposites. Multiply the numerator and denominator of the second fraction by $-1$ :

$\frac{2n^2 + 8n - 1}{n^2 - 1} - \frac{(n^2 - 7n - 1)(-1)}{(1 - n^2)(-1)}$

Step 2 — Simplify the second fraction. Multiplying gives $\frac{-n^2 + 7n + 1}{n^2 - 1}$ . Both fractions now share the denominator $n^2 - 1$ :

$\frac{2n^2 + 8n - 1}{n^2 - 1} - \frac{-n^2 + 7n + 1}{n^2 - 1}$

Step 3 — Subtract the numerators over the common denominator:

$\frac{2n^2 + 8n - 1 - (-n^2 + 7n + 1)}{n^2 - 1}$

Step 4 — Distribute the negative sign. Subtracting $(-n^2 + 7n + 1)$ reverses the signs: $-(-n^2) = +n^2$ , $-(+7n) = -7n$ , and $-(+1) = -1$ :

$\frac{2n^2 + 8n - 1 + n^2 - 7n - 1}{n^2 - 1}$

Step 5 — Combine like terms. Group the $n^2$ -terms: $2n^2 + n^2 = 3n^2$ . Group the $n$ -terms: $8n - 7n = n$ . Group the constants: $-1 - 1 = -2$ :

$\frac{3n^2 + n - 2}{n^2 - 1}$

Step 6 — Factor the numerator and denominator. The numerator $3n^2 + n - 2$ factors as $(3n - 2)(n + 1)$ , since the product is $-6$ and the sum is $1$ ( $3$ and $-2$ ). The denominator $n^2 - 1$ is a difference of squares: $(n - 1)(n + 1)$ :