Question

Please help me understand the zero factor property with this problem. I understand the property when the equations have parentheses, but not when they don’t. I know the answer I just need help explaining! Thank you!

Answer 1

Two ways to do this! There are some more but these are the most intuitive.

Way 1: Factoring

You can factor this equation into one with parentheses. A way I like thinking of doing this is by seeing if you can split the coefficient of the middle term into the sum of two numbers, which, when multiplied with each other, equal the third term. To see what I mean, look at the equation you gave.

`x^2 - 18x + 77 = 0`

`x^2 - 7x - 11x + 77 = 0`

We notice that when -7 and -11 are multiplied, they equal 77. This means we can factor the equation! As follows:

`(x-7)(x-11)=0`

If you want to see why this works, cross-multiply out
`(x-7)(x-11)`. You should see that it is equal to
`x^2 - 7x - 11x + 77`.

From here, you should be able to see the solutions are
`x = 7, 11`.

Way 2: Complete the Square

We can do something like this:

`x^2 - 18x + 77 = 0\\x^2 - 18x + 81 = 4\\(x-9)^2 = 4\\x-9 = \pm√(4)\\x = 9\pm 2\\x = 7, 11`

Essentially, what we are doing is we are adding / subtracting from both sides of the equation so the left side becomes a perfect square of some term
`(x-a)`. From there, we can just solve for
`x`.

Hope this helps you, please let me know if anything confuses you.