Simplify (u^2-4v^2)/(u^2+uv-2v^2)

$\frac{u^{2} - 4 v^{2}}{u^{2} + u v - 2 v^{2}}$

Step 1

Simplify the numerator.

Rewrite $4 v^{2}$ as ${(2 v)}^{2}$ .

$\frac{u^{2} - {(2 v)}^{2}}{u^{2} + u v - 2 v^{2}}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = u$ and $b = 2 v$ .

$\frac{(u + 2 v) (u - (2 v))}{u^{2} + u v - 2 v^{2}}$

Multiply $2$ by $- 1$ .

$\frac{(u + 2 v) (u - 2 v)}{u^{2} + u v - 2 v^{2}}$

Step 2

Simplify the denominator.

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 1 \cdot - 2 = - 2$ and whose sum is $b = 1$ .

Reorder terms.

$\frac{(u + 2 v) (u - 2 v)}{u^{2} - 2 v^{2} + u v}$

Reorder $- 2 v^{2}$ and $u v$ .

$\frac{(u + 2 v) (u - 2 v)}{u^{2} + u v - 2 v^{2}}$

Factor $1$ out of $u v$ .

$\frac{(u + 2 v) (u - 2 v)}{u^{2} + 1 (u v) - 2 v^{2}}$

Rewrite $1$ as $- 1$ plus $2$

$\frac{(u + 2 v) (u - 2 v)}{u^{2} + (- 1 + 2) (u v) - 2 v^{2}}$

Apply the distributive property.

$\frac{(u + 2 v) (u - 2 v)}{u^{2} - 1 (u v) + 2 (u v) - 2 v^{2}}$

Move parentheses.

$\frac{(u + 2 v) (u - 2 v)}{u^{2} - 1 u v + 2 u v - 2 v^{2}}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(u + 2 v) (u - 2 v)}{(u^{2} - 1 u v) + 2 u v - 2 v^{2}}$

Factor out the greatest common factor (GCF) from each group.

$\frac{(u + 2 v) (u - 2 v)}{u (u - 1 v) + 2 v (u - v)}$

Factor the polynomial by factoring out the greatest common factor, $u - 1 v$ .

$\frac{(u + 2 v) (u - 2 v)}{(u - 1 v) (u + 2 v)}$

Rewrite $- 1 v$ as $- v$ .

$\frac{(u + 2 v) (u - 2 v)}{(u - v) (u + 2 v)}$

Step 3

Cancel the common factor of $u + 2 v$ .

Cancel the common factor.

$\frac{(u + 2 v) (u - 2 v)}{(u - v) (u + 2 v)}$

Rewrite the expression.

$\frac{u - 2 v}{u - v}$

Simplify (u^2-4v^2)/(u^2+uv-2v^2)

Related Questions

Support LCMGCF.com by disabling your adblocker.