 Hello! Today we're going to be solving some quadratic equations and we're going to start with the square root property. Okay so let's go ahead and refresh some things about square roots. So I want to simplify the square root of 36. Now let me go to the calculator. So second my square root key is right here above my x squared so when I click that button that'll bring that up. Square root of 36 is 6. Okay this makes sense because so this is because 6 squared will give us 36. Okay now the square root of x squared I can't put this one in my calculator but hopefully you notice because what squared will give me x squared that's gotta be just a plain old x. Okay fantastic so what I wanted you to notice here then is that the square root undoes any kind of a perfect square. Okay so what we can do then is we can apply the square root to undo any squares. Now the key here also is to remember that a parabola is symmetric so we know we're going to end up with two answers. Not always but typically we'll have two answers with these problems. One answer is going to be on the right side of our vertex. The other answer is going to be on the left side of the vertex so one I'm calling positive one I'm calling negative depending on where your vertex is the signs may be a little bit different. Alright the square root property itself says that if x squared equals c then x equals plus or minus the square root of c. If c is negative then the equation has no real solution. Okay another key here is you need to isolate your squared quantity before you start applying any square roots. Okay to an example solve the equation 2x squared equals 6. Following this what I said here we have to isolate the squared quantity so that means I need to undo that multiplication by doing division so I'm going to divide both sides by 2 so that gives me x squared equals 3. Now to undo that square we're going to apply the square root so I'm going to do a square root on both sides and I have to remember one's positive one's negative. In this case our vertex is at zero so this makes perfect sense. Okay so that means x is going to equal plus or minus the square root of 3. Lovely answer. Okay if you don't like that answer let's go to the calculator and let's type in the square root of 3 and get an approximation. So my x's are going to be approximately 1.73 and then x is going to be approximately negative 1.73. So I got my plus and I got my minus. Whichever way you prefer to write your answer should be fine. Either way they mean exactly the same thing so it just depends on if you want an exact value or if you want an estimated value. Okay next one I want to again isolate my squared quantity so let me highlight that. So I need to undo that minus 7 that's sitting there and I'm going to undo that with a plus 7. So I end up with x plus 4 squared is going to equal 7. I want to undo that square with a square root so square root square root plus or minus. So that gives me x plus 4 equals another lovely number plus or minus the square root of 7. Okay not pretty but that's okay. So now I need to solve for x so I need to undo that plus 4 with a minus 4. And I'm going to stick that in front of the square root. The order there doesn't matter but it looks a little bit prettier. So in this case my exact answer is going to be negative 4 plus or minus the square root of 7. And that's a perfectly legitimate answer or let's go back to the calculator to get an approximation. So negative 4 plus the square root of 7. So second square root 7 hit enter. And then we can also do negative 4 minus the square root of 7. So my approximate values then are negative 1.35 negative 1.35 and negative 6.65. So those would be my two answers then an approximate value. Again doesn't matter which way you want to write them. Just depends on if you like exact values with ugliness or if you want decimals which is a different kind of ugliness. Okay fantastic. Next problem so this one has a little bit more going on negative 2 times the quantity p minus 5 squared plus 20 equals 4. Okay again I'm going to highlight this square right there. So we need to do some isolation. Let's undo this plus 20. So let's go ahead and subtract 20 from both sides. You could divide everything by negative 2 but then you'll have to remember to do it to each piece. So I think it's smarter to subtract 20 first. So that gives me negative 2 times the quantity p minus 5 squared and that's going to equal a negative 16, 4 minus 20. Now we can go ahead and divide by negative 2. It's a little bit easier with this one. So we end up with p minus 5 squared is going to equal 8. Okay now that I have my square by itself I can do the square root of both sides stick in my plus or minus. So I end up with p minus 5 equals plus or minus the square root of 8. Go ahead and add 5 to both sides. So I'm going to give an answer of p equals 5 plus or minus the square root of 8. Again if you want a decimal you can punch line in your calculator. I've shown you how to do that so hopefully you can handle that part of it on your own. Great. These are lovely answers aren't they? Okay next up. So my question is why would you want to solve quadratics this way? Well the good old Pythagorean theorem. You are going to need square roots to solve for the missing side. So if you remember from your trig days or maybe your geometry days the Pythagorean theorem says a squared plus b squared equals c squared. Okay where a and b are the legs of your triangle. So those are going to be your two sides that are perpendicular to each other. So these are your legs and c is your hypotenuse. So for many of these problems you're going to have applications. You're going to have to make sure to set up an accurate picture so that way you know if you've got a leg or you have a hypotenuse with whatever numbers are given to you. Okay so the next problem says to get from work to my home or sorry to get to work from my home I travel 10 miles north. Okay so let's label this. This is home. I'm going to travel 10 miles north then turn west that would be this direction and travel nine miles. So this is nine miles. If I could travel straight from my home to work. So straight. Wow that actually looks pretty good. Alright how much distance would I save? So ignoring my kind of sad looking drawing here hopefully you notice that this is actually a right triangle. The two pieces of information given are my legs and I'm looking for my hypotenuse. If I could travel straight so that would be my straight distance there. Okay so setting up my Pythagorean theorem that would give me 9 squared plus 10 squared equals c squared. So 9 squared is 81, 10 squared is 100, c squared we don't know yet. You add 181 that gives me 181 is going to be my value for c squared. Apply the square root. Now I'm not going to put a plus or minus on this one because a negative distance doesn't make any sense. So c is going to equal, let's go to the calculator, the square root of 181. So that is approximately 13.45. So I'll say this is about 13.45. Okay I haven't quite answered the question yet though because it says how much distance would I save? So typically I would go 10 miles north, 9 miles west. So that's a grand total of 19 miles minus that 13.45. So I would save, I believe that's 5.55 but let's double check. So 19 minus the answer that I just got. So I can do second answer and that gives me 5.5. My mental math worked. So I would save 5.55 miles and that would be my final answer to that problem. Alright last Pythagorean theorem problem. This is a ladder leaning against a building. Alright so let's go ahead and draw our building and pretend like that actually looks like a building. My art is not so good but my math is good thankfully. Alright we have a 30 foot ladder leaning against the building. Okay so there's my ladder. We said it's 30 feet. The base of the ladder is 10 or sorry 12 feet from the building. So this distance here is 12. And look at that I just made myself a right triangle. But this time I'm giving a hypotenuse and I'm giving one of my legs so I'm missing the other leg. And this question makes sense then because it says how high up the wall is the ladder. That's a leg of my triangle. Okay I'm calling that missing distance A. You could call it B. Doesn't really matter. So A squared plus 12 squared equals 30 squared. Got an extra zero in there. A squared plus 144 equals 900. If you don't know those values just plug them into your calculator. Subtract 144 from each side. So A squared is mental math. I believe that's 756. Do the square root of both sides. Square root square root. Again we don't need plus or minus because this distance has to be positive. So let's go to the calculator 7. Square root of 756 gives us a total of 27.5. So A is 27.5 and this would be in feet. Okay a good gut check that this answer makes sense is it has to be smaller than your hypotenuse. So is 27.5 smaller than 30? Absolutely. So that's a good gut check. All right. Fantastic. More quadratics to come.