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Determine the number of solutions to each quadratic equation. (a) $4 x^{2}-5 x+16=0$ $36 y^{2}+36 y+9=0$ $6 m^{2}+3 m-5=0$ $18 n^{2}-7 n+3=0$

Welcome. We're here today to determine the number. Now we will add to both sides of our expression to complete the square so that we can continue on by solving this equation. Note we will not be solving the equation in this video. I highly recommend that you use a six step process outlined on page 677 of your textbook to aid you through this process. Step one asks us to verify that the coefficient of our second degree term is one. As you can see, here are coefficient of our second to return is actually four. So in orderto rectify this situation, Step one tells us to divide our entire equation by the coefficient off our second to return. So we're gonna divide this entire equation by four rewriting our equation. Then we get Z squared. Z squared minus Z over four Z over four is your four minus 39/4 equals zero equals zero. So going back to step one, we can now verify that the coefficient of our second degree term is, in fact one and step two toes is to then move all constance to the right hand side of our equation rewriting our equation. Then we get Z squared. G square. Mine is your for minus Z Over four equals 39/4. 39/4. Perfect. Now, Justus, you did and exercises three through 11 in section 11.2. We will go ahead and find the value to complete the square. To do so, we must identify the coefficient of our first degree term, which, in our case, is going to be negative. 1/4, uh, taking 1/2 of negative 1/4, which is also known as multiplying negative 1/4 by 1/2 some multiplying. Negative 1/4 by 1/2. We get negative. 18 Negative. 1/8. As you know, the standard form of a quadratic is a Z squared, plus B Z plus Z equals zero. So what we're actually doing Let me just put the general form here. We're taking half of B and were squaring it to complete the square. So continuing on with that calculation, we're gonna square negative 1/8 to find 1 64th 1 64th So I'm gonna box in blue. This is the value. This is the number that we have determined that will complete the square. So writing our final expression, we have Z square Z squared. Mine is your for on a Z over four plus 1/64. 1/64 equals 39. What? We're for over four plus 1/64 plus 1/64. I hope this helps and keep at the great work.

Okay, so first steps, plug in system in your equations into matrix. Negative for six. First room six ***. 12. 15 nine. I think you ate 14. 90 Andi. Negative age. Okay, Now I'm just gonna reduce this matrix and then solve for X one x two extremes. So the first step is to divide first wrote for luck. It's one negative too negative. Five to do to first. True on the second to and two other rose will stay. Seem so six negative. 12 negative. 15 nine. Negative age for Gene. 19 and innovate. Okay, now we're just gonna go ahead. Eliminate first call. That's gonna give us one, Teoh. Sure. Two through five. Over two. 1/2. Zero on three. Over two. Negative to ju. Now we're gonna second columns 10 to June. Negative to to 1/2. You to. And you have to, don't you? This It gives us a solution. Thanks. One to negative. 5/2. 1st 3/2 x three. Next to physical. Too negative. Too minus one to next three. Where X tree is free. Variable

Okay, so we want to determine the number we need to complete North, where for our first step would be to move our constant to our right hand side. And then we also need to factor out this war to have a coefficient of one for the squared. So that's going to before. And then here's the Squared minus 1/4 D plus something, and that's equal to 39. Close something. Now let's find that Constance that's going to be over to your power to So what's B? That's a negative one over for that 2nd 1/4 times to which is eight to hurt, too, and that's equal to 1/64.

First step is going to have to plug in the system of linear equations into the Matrix. So you get a little too for matrix Negative. 38 Negative. 12. Negative. Six, 924 Negative six. Okay, now we're just gonna compute this matrix. Apply grow operations to reduce it down. First step is going to be to make the pivot in the first color by dividing the first room for two 14 Negative. 32 second rose. Just stay sticks. Okay? Now, I could just eliminate the first column since this is a second Roosevelt full in the first row. So get 14 on four, three to on. Cheers. Since this religious three times this truth you don't get No, But you know that this sex one, this is six 263 That is be right, x one as to punish for thanks to those three x three where it's too on X three free variables