SM2 9.2-5: Solve for a Quadratic Given the Length of part of a Diagonal and the Whole Diagonal

追加: 2026-05-26

[00:00:00]Hello and welcome. So we're told that given a parallelogram ABCD, right? We're told that BE, the segment BE, is 12x plus 96. BD is 6x squared.  So first off, B to E is going to be this little segment right here. So that's 12x plus 96. Then we're told that BD, which is not just part of the segment but the whole line itself, is going to  be 6x squared. So your first question and when anything geometry related it's always going to be, all right, what's the relationship? How does BE and BD relate to each other? Well we know there is a particular property with parallelograms that when you have two of your diagonals, right, and when they meet this point E, we call that the midpoint, okay? It's going to meet directly in the middle of both of these lines. That is the direct middle of both lines, which means that BE and ED, that segment, those are congruent to each other.

[00:01:07]So if this section was uh 17 this section would also be 17 and the whole thing would be 17 plus 17 which is 34. Okay. And the whole segment would be 34 altogether. All right. And that's going to be true no matter what you have. Which in this case we're told that BE is 12x plus 96, which means that the other section ED must also be 12x plus 96. And we're told that the whole thing should be 6x squared. But we also know that it's  more than just 6x squared. We also know that it's a 12x plus 96 and the 12x plus 96  combined. So if you take 2 of our 12x plus 96's that should be the exact same thing as 6x squared, okay? Another way you could set this up, okay, an  equivalent way to set this up is you could say, okay, well we know that it's two of the  small portions to give you the big one.

[00:02:14]Or you can take half of the large portion of  our blue line BD, and that should equal one of these smaller portions, okay? Either  or, these are both totally correct setups, okay? Whatever floats your boat, go for  it. It does not matter which way you go, okay? So I'm just gonna go with this initial setup  that we have over here. So I'm going to multiply by two. All right. So I'd say, all right, so then  we have 24x plus 96 times 2, which is 192. Equals 6x squared. Now you just need to solve for your x, right? Hopefully  as soon as you see that x squared you should be getting like bells ringing like, oh wait, yeah, I  need to solve for our x squared. And the only way you can do that is one of three ways. You  can either factor, use quadratic formula, or complete the square. I like factoring. We're going to use that, okay? So I'm going to subtract the 24x as well as subtracting the 192. Now the reason why is I like my squares being positive, so that's why I'm moving everything to the  right side. Couldn't you move the 6x squared to the left by subtracting it? Heck yeah.  All right. Will you get the same answer? Yes. Okay.

[00:03:39]And then from there you just solve. So if we're solving by factoring my first step would be to take a GCF which is 6 in this case. So we have 0 equals 6 parentheses x squared minus 4x. 192 divided by 6 is 32. And now you will just finish factoring  from there. So your middle number, negative 4, your first one times negative 32 is negative 32.  Now there's only two numbers that is going to work here. That is going to be 8 and 4. To get to a  negative 32 we're going to have a negative 8. So negative 8 plus 4, negative 4, and then negative 8 times 4 is negative 32. Those are your numbers.

[00:04:23]So then we have 0 equals 6 parentheses x minus 8, x plus 4. We can go straight into those factors only because we have a positive 1 in front.  If it wasn't you have to factor the long way, okay? And then from there you'd say,  all right, great. Now we have our two factors. We have x minus 8 set it  to zero, x plus 4 set that to zero.

[00:04:48]And then you can solve your equations. That's like a squish zero right there. There we go.

[00:04:53]So the first one you would add 8, x equals  8. Next one subtract 4, x is negative 4.

[00:05:00]Right? You wouldn't have to worry about the 6,  that doesn't make a solution because there's no x attached to it. So then you have two solutions. You have 8 and negative 4.And then from there all you'd have to do is you just have to double check like plug them in. And when you do you're gonna end up getting maybe one of them fails, maybe one of them doesn't. If they both fail then you wouldn't have a solution, right? So first off  let's plug it in. So first let's try x is 8, okay? So if we do 12-- whoops, 12 times 8 we're going to get 96 plus another 96 we get 192. When we plug 8 in for our x there. We plug 8 in here we get 8  squared which is 64 times 6 is going to get 384.

[00:05:51]Which they're both positive numbers. Awesome, right? Distances can't be negative, which means that 8 is a valid solution, okay? Now the length of BD we just found that when x is 8, 384 was the number we got back. Huzzah! Okay. Now let's try negative 4. Let's make sure that that still works as well.

[00:06:14]So now we're going to try negative 4 in  instead. And when we do we get 12 times negative 4 that's going to be negative 48  plus 96 is 48. Okay. Now plug negative 4 into our next one. So we have 6 times negative  4 squared. Negative 4 squared is 16. All right, negative times a negative is a positive. 6 times 16 is then 96. Both positive numbers which means that negative 4 also is a valid solution. And  then when we plugged into BD we got 96 back. So you have two possible lengths for BD. If x is 8, that length is going to be 384. If x is negative 4, then 96 will be the length that we get back,  okay? It's an either/or situation. They're not both going to be true at the same time but you  have two cases, right? It's one or the other.

[00:07:09]And that would be it. You'll be done. That'd  be your answer. Thanks so much for watching.