SM2 13.6-5: Solving a Non Factorable Quadratic with Two Secant Lines

Hinzugefügt: 2026-05-09

[00:00:01]Hello and welcome. So we are given a circle  with two secant lines. All that means is that you have a line, right? Two of those lines  intersecting each other at point A and that line in particular intersects the circle at two  places. Now it's that double intersection is why we call it a secant line. That's literally all it  means. It just means that you're intersecting the circle at two places. All right. Now if you had  it only intersecting the circle once that would be what we call a tangent line. All right, so  don't worry too much about that vocab there.

[00:00:32]All right, so if we have two secant lines we have  the intersection. All we got to do is know the relationship between them. Well that relationship  will be as follows: you'll have the part times the whole equals part times the whole. Is the  more or less bare minimum you need to remember, right? With part, we are specifically referring  to the-- as the outside of the part. So for example, we have uh three dots, right? The initial dot is  your point A and the first part is going to be to your first intersection point. So that means  that 2r plus 9 is going to be our outside part, right? It's outside of the circle. It's that  first part. Now the whole is going to refer to the whole line, right? The entire from the  initial point A to point G, right? The second intersection point. That is the whole. Now you  got to be careful. That's not just 2. That's 2 plus the 2r plus 9. That is the entire section  right there. Okay. And then it's just a repeat like rinse and scrub for the next one. So  the first part is 6. So part 6. The whole is not just 3. It's 6 plus 3, right? It's the  entire section right there. And that would be it. That would be your setup for this one. Okay.  And then it's just a matter, let's multiply it together. Solve for your r, right? And then we  have 2r plus 9 first parentheses and 2r plus 11 second parentheses, right? 2 plus 9 is 11. And  then equals 6 plus 3 is 9 and 9 times 6, 54.

[00:02:15]Okay. And now it's just a matter of solving  for r. Now you might be tempted to set these parentheses to zero but that only applies if your  equation equals zero. It doesn't. Which means you have to distribute these parentheses out. So  2r times 2r, 4r squared. 2r times 11, 22r.

[00:02:34]9 times 2r, that'll be 18r. 9 times 11, 99. Okay.  Now just combine your like terms. So we have the 22 plus 18 that'll be 40. So we have 4r squared  plus 40r. And then we'll do, I'll just move the 54 while we're at it. 99 minus 54 is going to leave  us with 45. Whoops, as I write 5 first. There we go. 45.

[00:03:03]Okay. Now I'm going to save you a little bit  of time. Normally at this point you would be solving a quadratic. Okay, now quadratics are  normally factorable. Okay, a good a good chunk of them are. This particular quadratic for the  sake of time in this video I'm just going to tell you right now it isn't factorable. All right.  Which means that we got to work outside the box a little bit here. And I'd say all right well what  are your other solving methods? You have either completing the square or quadratic formula. Which  I personally prefer the quadratic formula if I had to pick between those two. So then I'd say right  so that means your quadratic formula variables are here. So r equals negative b plus or minus  the square root b squared minus 4ac all over 2a.

[00:03:46]Remember a is your coefficient out front. b  is your coefficient in the middle. c is your coefficient at the end if you're in standard form.  All right. So then it's just a matter, filling in the numbers. So we have negative 40 plus  or minus the square root of b which is our 40, whoops, squared minus 4 times our a, which is 4, times  our c, which is 45, and that is all over 2 times a which is 4. Okay. Save a little time. I'm  going to punch the numbers in a calculator.

[00:04:22]We have negative 40 plus or minus, and this whole  section is going to leave us with 880, and then all over 8. Okay. At this point read your directions,  it says give your answer rounded to three decimal places. Meaning do we need to simplify this to  get an exact answer? No, right? If it's asking for a decimal you don't need to simplify this.  Now save yourself some time, in which case plug it right into a calculator, right? So that then  I've already plugged in a few of these for us.

[00:04:54]Okay. You'll notice that up top we have negative  40 plus the square root of 880 over 8. Down below we have negative 40 minus the square root of  880 over 8. Right. And those two answers are negative 1.29 and, sorry, negative 1.292 and negative 8.708. Now those are both negative numbers but they're not an issue, right? They're not both an issue because  remember you're solving for r, right? Where is r?

[00:05:25]The segment from A to Y and that is a length  and lengths need to be positive. So if you can plug in your answers, right? Those two negative  decimals and get a positive number back they're fine. They're good solutions to use. But if  they're negative that's an issue. Also plug those in right here for you. So the first one we  can see 2 times negative 1.292 plus 9 and we end up with 6.416. That is a positive number. It's  good to go. That means that you have at least one solution. Okay. Now your other one comes back as a  negative 8.416. That's an issue, right? Can't have negative lengths which means that that solution  kicks the bucket. And your final answer is going to be r equals negative 1.292. And that's it.  Okay. It's really just a matter of knowing your relationship. Set it up right, and factor if you  can. If not use quadratic formula, and then you'd be good to go, right? Make sure it comes back  positive. And that was it. Thanks for watching.