 In this video we provide the solution to question number 12 from practice exam number 2 for math 1050 in which case we're given a quadratic equation and this situation is 3x squared plus 2x minus 2 that's equal to 0 And we have to solve this quadratic equation by completing the square and we have to find all complex solutions It could be real they might not be it doesn't matter We have to find all of the solutions even if they're non real and the important thing on this one Is we must complete the square the instructions say so if we try to solve this by factoring Which really won't be helpful in this one. We get no credit if we use the quadratic formula Even if our answer is correct the instructions say complete the square So if we do it some other way that doesn't count for anything So we need to make sure we complete the square here now before we start completing the square We need to make sure this isn't the standard quadratic form the right-hand side should equal zero and then the left-hand side We should combine any like terms if that hasn't already been done. That's what we're in the standard quadratic form. We're ready to go So what we then need to do is we need to separate basically the constant From the variables there. So I said set equals zero. Well, JK. We're gonna move the two to the other side So we have a 3x squared plus a 2x and this is equal then to 2. You'll notice there's a gap here This is our We're waiting for our guest of honor. This is his seat that we left open. We want to go from there So then looking at the we should just have multiples of x on the left-hand side We have to multiple our factor out the multiple of three from the leading coefficient So we take out the three that leaves behind an x squared now two is not divisible boy three as a whole number But it is as a fraction So we still have the factor out that three away and that leaves behind two-thirds x and this is equal to two right here So now we're in the position where we want to start calculating who is that guest of honor So who goes in this seat right here? Remember how we do that we look at the middle coefficient after the leading coefficient's been factor factored away We have to take half of that thing So one half of two-thirds of course is equal to two-sixth or one-third like so then we have to square it We square this thing and we end up with one ninth That's the guest of honor that goes in right here one ninth But what's good for the goose is good for the gander if we add one ninth to the left-hand side We have to do that to the right-hand side as well, but we really didn't add one ninth We've added three times one ninth because that three would distribute on to the one ninth as well So make sure to write the coefficient of three attach the one ninth as well Then from there we get three times well inside the parentheses you have x squared plus two-thirds x Plus one ninth. I should have put a plus sign right there This is now a perfect square trinomial Factor and it'll factor to be x plus one-third that is half of the two-thirds we saw right there quantity squared If this is a plus this is a plus if this was a minus then this would also be minus the signs match in that regard So the left-hand side is now factored we get a two plus well three times one ninth is one-third like so And so now we want to then continue to solve the equation All right now that we got x all by itself We have to peel the onion and remove everything that's attached to the x so to do that We're gonna divide both sides by three like so also since I have to add one-third with two I should probably write the two as a six-thirds like so so notice on the right-hand side You get six-thirds plus a third that's gonna give you seven-thirds We didn't divide it by three, which is the same thing as times you by one-third so that gives us a seven-ninths right there So then where are we in the process now? We get x plus a third squared is equal to seven-ninths like so We then want to take the square root of both sides to get rid of this square on the left-hand side But remember Squaring is not a one-to-one function So its inverse actually has two possibilities the plus or minus there on the left-hand side the square in the square We will cancel out giving us x plus a third on the right-hand side We then have this plus or minus the square of seven over the square of three Excuse me the square of nine little head of myself that the square of nine is three So we get the square of seven over three like so and then to finish up here We're gonna subtract one-third from both sides of the equation Notice they do have a common denominator three which makes things a little bit easier for us And so then the final step here is just to record the answer We're gonna get x equals we have negative one-third plus or minus root seven over three So we'll just write this as negative one plus or minus the square of seven over three If you want to separate your answers into two pieces negative one plus root seven over three and negative one minus root seven over three You can do that, but we want to keep our answers exact and because the square of seven is this irrational number We really can't simplify it any more than just approximating it So honestly if you want to leave the two answers together with a plus or minus symbol That's perfectly fine. No issue with that whatsoever And so we record the answer as x is negative one plus or minus root seven over three This would be the same answer we got from the combat the quadratic formula But like I said as the instructions required we had to solve this by completing the square