Question

Factor.
(3y+2z)² - (3y-2z)²

Answer 1

`(3\, y + 2\, z)^(2) - (3\, y - 2\, z)^(2) = 24\, y\, z`

Explanation:

Make use of the fact that for any
`a` and
`b`:

`\begin{aligned}& (a + b)\, (a - b) \\ =\; & a^(2) - a\, b + a\, b - b^(2) \\ =\; & a^(2) - b^(2)\end{aligned}`.

In other words, the difference
`(a^(2) - b^(2))` between two squares could be written as the product of
`(a + b)` and
`(a - b)`.

Apply this identity to rewrite and simplify the expression in this question. In this example,
`a = 3\, y + 2\, z` whereas
`b = 3\, y - 2\, z`.

`\begin{aligned} & \underbrace{(3\, y + 2\, z)^(2)}_(a^(2)) - \underbrace{(3\, y - 2\, z)^(2)}_(b^(2)) \\ =\; & \underbrace{((3\, y + 2\, z) + (3\, y - 2\, z))}_((a + b))\\ &* \underbrace{((3\, y + 2\, z) - (3\, y - 2\, z))}_((a - b)) \\ =\; &6\, y\, (3\, y + 2\, z - 3\, y + 2\, z) \\ =\; & 24\, y \, z \end{aligned}`.