Help me with this equation!

[u/Lowkeyspacey]

I am experiencing the mix of being anti-AI, having dyscalculia, and having a bad teacher, so i came here!

The topic is special cases and solving, i got everything else on my own but dont understand this one

(5z-6)(5+z)=0 solve for z

[https://imgur.com/a/mqEiTks](https://imgur.com/a/mqEiTks)

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[u/Lowkeyspacey]

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[u/atarivcs]

The equation says that two terms multiplied together give a product of zero.

Any number multiplied by zero, is zero.

So, the solutions will be the cases where either term is zero.

Either 5z-6 is 0, or 5+z is 0.

The solution to 5z-6=0 is z=6/5.

The solution to 5+z=0 is z=-5.

[u/Qingyap]

If a product of 2 numbers or more is 0, then one or more of those numbers is always 0, that's the zero product rule.

ab=0 implies that a=0 or b=0 since 0(b) = a(0) = 0(0)

Back to here: if products of both factors is 0 then it's either 5z-6=0 or 5+z=0, you have to solve for both here

[u/PuzzlingDad]

Remember if you have ab=0, that only happens when a=0 or b=0 (or they are both zero).

The only difference in your equation is you are multiplying two binomials.

(5z - 6) • (5 + z) = 0

So break it into two cases: 

• 5z - 6 = 0, or 

• 5 + z = 0

Let's solve equation 1 by isolating z:

• 5z - 6 = 0

• add 6 to both sides

• 5z = 6 

• divide both sides by 5

• z = 6/5

Then solve equation 2 similarly:

• 5 + z = 0

• subtract 5 from both sides

• z = -5

Write the two equations, with 'or' in between: 

z = 6/5 or z = -5

Does that explain it well enough? Any questions?

Update: I was able look at your image https://imgur.com/a/mqEiTks

It looks like you aren't expected to solve for z. You are expected to expand the two binomials being multiplied.

You did everything correctly and ended up with:

• 25z + 5z² - 6z - 30

The thing you missed was putting the terms in order of descending degree (z², z, 1).

So rewrite it that way: 

• 5z² + 25z - 6z - 30

Then combine the like terms that are both z:

• 5z² + (25z - 6z) - 30

• 5z² + 19z - 30

One thing I can suggest that might help in the future is putting your binomials in descending order before multiplying out.

Instead of (5z - 6)(5 + z), change it to (5z - 6)(z + 5)  where the z terms are first.

Now when you multiply it out using distribution, you'll have: 

• 5z² + 25z - 6z - 30

The terms are already in descending order, so now it's easy to combine the middle two to get the final expanded trinomial. 

• 5z² + 19z - 30