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Solve each quadratic equation by the method of your choice.$$3 x^{2}-4 x=4$$...

Question

Solve each quadratic equation by the method of your choice.$$3 x^{2}-4 x=4$$

Solve each quadratic equation by the method of your choice. $$3 x^{2}-4 x=4$$



Answers

Solve each equation by factoring.
$4 x^{2}=-3 x$

Okay, So this question here asks us to solve this equation by using the quadratic formula. Now, I'm not a big fan of being told how to solve something, because I can quickly factor this and say Okay, well, this is minus three and X minus one. And so my solutions are one comma three. And so I you know, that's ah kind of my opinion about this, but to answer this question will go through Ah, the rigmarole of the quadratic formula. So we know that it is a minus B plus or minus the square root of B squared minus four. A c divided by two A. So here I have a negative four. So a negative b would be a positive four plus and minus negative Four squared is 16 a minus, four times a is one. So I just leave that off in time C is three, and then divide that by two times a day. And again a was one, so we can just leave that, and then we get four plus and minus. This is the room for over to, and then this is gonna equal four plus and minus two over to well, you can reduce this. And remember, when you have a plus and minus in here, you have to reduce both. So I get a two and a one, so this is equal to two plus and minus one. So I always start with the plus cause if you do it an order from top to bottom and just make sure you don't mess up and do it wrong. So here two plus one is three and two minus one is once we get the same answers that we got by factoring, um, just in about three times the amount of time. So there you have it.

To complete the square. We take the B term divided by two square it and we end up with 9/4. Now, given this, we know that we can substitute this and add it on to both sides. So X squared minus three acts plus nine divided by four equals 4%. But before so I guess I we're adding nine forts to both sides. This gives us axeman is three divided by two because remember the square root of nine forces three divide by two equals 25 Divide by four. Factoring this we end up with axe equals three over two. Let's remind us five divided by two, giving us two solutions eight over to and negative to over two, which is equivalent to four and negative one.

To solve this equation, using the quadratic formula so it's already set up so that we can just plug it in. So we have negative B plus or minus the square root of B squared, minus four a c over to a. So first we have a negative B, since B is negative, for that turns that into a positive and then B squared is negative. Four squared, minus four tonnes. A, which is three and times C, which is a negative, too. Over two times A, which is three. And then we're left with a four plus and minus. So four square to 16 and a four times of three's 12 times a two is 24. We have a negative and a negative, so that turns that positive. So this will be a positive 24 2 times three is sticks. So now I have four plus and minus the square root of 40 over six. So now what we want to do is take this 40 and reduce it, and we can reduce that 40 to be the root of four times the root of 10 through the forest to so this becomes to Route 10 over six, and now we see We can reduce thes So we just divide each one of these by two and we're end up with two plus and minus the route 10 over three. And because this er is not a perfect square underneath here, we can just leave our answers in this form.

We're going to solve this problem by using completing the square. And so the first thing that I see here is that I'm gonna want to get rid of this four, that up in front of this x squared so I could just divide everything by four. And when I do that, I get X squared plus acts plus something, and we're gonna figure that part out next, divided by four, is equal to three force. And then we're gonna take be so be, is one Divide that by two and square it, and that's gonna be the number that we under here, which is 14 So we'll add 1/4. And when I add 1/4 to the left side, I got to do the same to the right, and you can see that this becomes one here because three forced plus 1/4 is four over four, which is the same as one. So next we have this here, so X plus 1/2 squared is equal to one. Then we're gonna take the square root of both sides. Remember to put that plus and minus in there and then I get X plus 1/2 is equal to plus and minus one cause. The square root of one is just one and then assuming a subtract 1/2 from both sides. So I get that X is equal Thio negative 1/2 plus and minus one. And in this case, we are going to come by these because we can, and we're gonna rewrite one as to over two. So now I'm gonna start with a negative 1/2 plus two over two. So that's equal to 1/2 a positive 1/2 because this is a negative 1/2 and then plus two over to as a positive 1/2 and then I have a negative 1/2 minus two over too. So my other answer here is minus three over two. So these are my two answers here for this quadratic equation.


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