Point) Solve the following equation for p=42p + p2 _ 9 p + 36= 0 p -3p =2Help: If there is more than one correct answer; enter your answers as a comma separated lis...

Complete each solution to solve the equation. $$ \begin{aligned} &\begin{array}{r} {p^{2}-p-6=0} \\ {(-3)(p+2)=0} \end{array}\\ &=0 \quad \text { or } \quad p+2=\\ &p=\ \quad | \quad p= \end{aligned} $$

All right. We have negative for times. The quantity p minus the quantity. Three minus T equals three kinds. The quantity six p minus two. So now I'm gonna start with my inner parentheses, and I'm going to distribute that negative sign. So I have p minus three plus p equals, and I'm distributing the three. 18 p minus six. I'm going to add my piece together, so I have negative for times two p minus three equals 18 p minus six. This gives me a negative eight p plus 12 equals 18 p minus six. Subtract 12. Subtract 12. Uh, I put them in the wrong column minus 12. It's minus 18 minus 18 p minus 18 p. So I have negative. 26 p equals negative 18. Divide by negative 26. No, By negative, 26 p equals 18/26 and two goes in 18 9 times and two goes into 26 13 times. So 9/30

Have three minus 7/2. P plus two is equal to six over the quantity to P plus two square. So what we can do here is well, multiply both sides of our question by well, by the LCD, which is to P plus two squared. So if we do that we get is well, we get three times to pee, plus two quantity squared rights are multiplying both sides by two people quantity to P plus two squared. That's gonna distribute. Give me a three times two people Esther squared and then a minus or minus just seven times two p plus two. Right, Because one of the factors in a cancel leaving just one factor of two people us to. And that's gonna be that equal to just six. Right? Because to people's to square over to pick up to over two p cluster square cancels out, there's going to be equal to six. And if I move the six for the other side of the equation, we're gonna get minus six is then equal to zero. Now, notice this might not look like, you know, the most nicest equation sort of south for what piers. But notice that if what we have to pee plus to write occurring twice, you have occurring once to the second power and then once here to the first power. So if that was just equal to some variable, well, then this will be just a nice quadratic equation of South. So let's still we let a variable Let's say we let you be equal to to pee plus two. Okay, then what we get is just while three times you square you just get three you squared and then minus will just seven times you so minus seven you and then minus six is equal to zero. Okay, so now well, looking at this, I can this be factored Well, looking at it right now, um, it looks like it wants to be factored. Well, what I can actually do here is that can really let this not be changing much. I could rewrite this as three you squared, and instead of minus seven, you I could write it as minus nine. You, But then plus two, you just mind seven. You then minus six is equal to zero. If we do this, then we can know we can actually factor this by grouping, I can factor these 1st 2 terms. I can factor out a common factor of three. You. So you three you times while you minus three. Right? And I am factor out a two out of these. Last, um, to this as you should be a two u two plus two u minus, X factor out a common factor of two out of to you and minus sex like a plus two times U minus three. Is that equal Touchstone zero? Or now I have a common factor of U minus three. I can go ahead and pull out a common factor of U minus three. So we get U minus three. Well, then times through you, but get back to our first term. So times three you and then a plus two okay, and then this is still equal to zero Also. Now what we have is we have two factors being multiplied together. That's equal to zero. So this tells us that either u minus three has to be equal to zero or three. U plus two has to be equal to zero. Well, if you minus three is equal to zero, that implies that you is equal to three. Right? But if three U plus two is equal to zero, then we have what three u is equal to negative two. So therefore you is equal to negative two thirds. So we have you is equal to three or us equal to negative 2/3. Now, we weren't actually solving for you when we saw for you here. We're trying to sell for originally trying to sell for Pete. But remember, we let you be equal to two p plus two. So if we have you, it's equal to three. Well, then we know that two p plus two is equal to create right, and it's two p plus two is equal three. That means that two P is equal to three months to or one. So therefore, API, is he gonna want about by two or 1/2? Okay. And then we have or will. You could be equal to negative 2/3 off you if you go to native 2/3. That implies that two p plus two which is equal to you right, is equal to negative 2/3. Okay, well, this implies that two p is equal to while to, um, or while negative two minus 2/3 right. Which implies that P is equal to negative 8/6, which is equal to negative 4/3. So we have P is equal to 1/2 or P is equal to negative 4/3. Okay, so then have to go ahead and this check our solutions and make sure you're actually valid solutions for P No, we actually made a substitution, and we solved for you on back substitutes we could ever have after an extraneous solution here. So it's a check both the solutions. So first, let's go ahead and check. Um, 1/2. So if P equals 1/2 remember, original equation was three minus 7/2 plus three minus 7/2 peoples to equal six over the quantity to P plus two squared. So substituting in peak was one. After we get that three, um, minus 7/2 times, 1/2 plus two is going to be equal to six over, while the quantity two times 1/2 plus two squared OK, when we get here, is that we get that Well, three minus 7/3 is equal. Teoh 6/9 which implies that while 2/3 my or 2/3 equals 2/3 right, Because this supervised 2/3 and so does this. So we get at 2/3 is equal 2 2/3 which is true, right? 2/3 equals to 2/3. So therefore, um, he equals 1/2 is a ballot solution Now are other solution was p equals negative. 4/3 the same where we plug in negative 4/3 the check. Now we check three minus seven over two times negative, 4/3 plus two. And that should be equal to six. Divided by to times negative. 4/3. Uh, plus two. And that whole thing, we're being square. Well, here we end up with three plus 21 halves from over two is equal to 54 over four. Okay, well, this implies that 27 halves is equal to 27 halves. If we reduce this 27 have is equal to 27 halves, which again, is a true statement. So both of our solutions are a ballot solutions. So therefore, our solution set is the set containing negative for thirds and one. All right, take care

We'll start off by combining like terms. Nine p minus four p is five p and on the right side, seven p minus three p is four p. Next, we will subtract four p from both sides that way, all over peas are on the same side. Five p minus four p is one p plus six. Equal six. Next weekend. Subtract six from both sides, thes cancel out and six minus six that also cancel out to make zero, so P is.

Way won't solve this equation. Find out. Are you Pete satisfies this East. We do this. Get a lot of variables on one side on. Well, Constance on the other. So take three people. Both sides will have people. Us, too, is equal to minus seven. And puts were conference over here, so I might have to also site will have P easy concert minus nine on dhe doesn't satisfy our equation. Top. Well, four times minus nine from minus 36. Plus tip is minus there before on three. Test by seven is minus three times minus nine is minus 27. My 37 months seven is my state for so. Yet it does so P C minus nine. Thank you.