How do you simplify (3-2 sqrt 7)(3+2 sqrt 7)?

Multiply the quantities together piece by piece and get to -19 as the answer

Explanation:

The problem here has the form of (and excuse me as I use non-math teacher language)

(number - yucky thing) * (same number + same yucky thing)

The thing to realize is that when we multiply the number in the first quantity by the yucky thing in the second quantity, it will be neutralized by the number in the second quantity multiplied by the yucky thing in the first.

Let's do this piece by piece:

$3 \cdot 3 = 9$

$3 \cdot 2 \sqrt{7} = 6 \sqrt{7}$

$- 2 \sqrt{7} \cdot 3 = - 6 \sqrt{7}$

$- 2 \sqrt{7} \cdot 2 \sqrt{7} = \left(- 2 \cdot 2\right) \cdot \left(\sqrt{7} \cdot \sqrt{7}\right) = - 4 \cdot 7 = - 28$

Add the 9 and the -28 together to get -19. The two terms with the square roots add together and become 0, so the final answer is -19.