Question

NO LINKS!!!!

The equation you wrote in part (c) of problem 8-65 is called a quadratic equation.
(2x^2 + 5x - 12=0) To solve it, you need to examine what you know about zero. Study the special properties of zero below.

Nathan, Sonia, and Gaston, are playing a game where Nathan and Sonia each think of a number and then give Gaston a clue about their numbers. Using the clue, clue Gaston must tell them everything that he knows about their numbers.

a. Nathan and Sonia's first clue for Gaston is that when you think to multiply their numbers together, the result is zero. What conclusion can Gaston make?

b. Disappointed that Gaston came so close to figuring out their numbers, Nathan and Nancy invite Nadia over to make things harder. Nathan, Nancy, and Nadia all think of secret numbers are multiplied together, and the answer is zero. What can Gaston conclude this time?

c. Does it matter how many numbers are multiplied? If the product is zero, what do you know about one of the numbers? This property is called Zero Product Property. Write a description of this property below.

Answer 1

Part A

Either Nathan picked 0, or Sonia picked 0, or both.

This is because multiplying nonzero numbers together gets a nonzero result. So one of them must have picked 0.

==============================================================

Part B

It's the same idea as part A. It's not clear what the nonzero values are, but one (or more) person picked 0 as their secret number.

If they picked something like 1, 2 and 3, then the product is 1*2*3 = 6 which is nonzero and the product is larger than the three original values. This is because each value is 1 or larger. If someone picked a small decimal value like 0.1 then 0.1*2*3 = 0.6 is the product. It's closer to 0, but not 0 itself.

==============================================================

Part C

Zero Product Property:

If m*n = 0, then either m = 0 or n = 0 or both.

This idea says that if the product of two numbers is 0, then at least one of the numbers must be 0 itself. It can be extended to three or more numbers.

This idea is useful when it comes to solving factored quadratic equations.

In the case of 2x^2 + 5x - 12, it factors to (2x-3)(x+4)

So using the zero product property, we can solve the quadratic equation like this

2x^2 + 5x - 12 = 0

(2x-3)(x+4) = 0

2x-3 = 0 or x+4 = 0

2x = 3 or x = -4

x = 3/2 = 1.5 or x = -4

The use of the zero product property happens in step 3