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Solve the following.$$7 x^{2}+2 x-3=6 x(x+4)+x^{2}$$...

Question

Solve the following.$$7 x^{2}+2 x-3=6 x(x+4)+x^{2}$$

Solve the following. $$ 7 x^{2}+2 x-3=6 x(x+4)+x^{2} $$



Answers

Solve the following. $$ 7 x^{2}+2 x-3=6 x(x+4)+x^{2} $$

To solve this rational equation. We usually want to clear out the denominators first. To do that, we're going to multiply by the L C D. To identify the LCD, we need to make sure that every denominator is fully factored. So looking at this denominator here, X squared minus five X plus six. We're gonna want a fact with that. And we see that negative three and negative two multiplied a positive six, but add to negative five. So these were the factors of that denominator. Bring everything else down. All right. We can now identify the LCD. The lowest common denominator needs toe have one of each expression or factor, that is in all of the denominators. So X minus three. And there's an X minus two, and that covers all of the factors air expressions that are in each denominator. So we're gonna multiply everything by the LCD here. Okay, Now, when we multiply this first fraction by the LCD, the denominator is going to cross reduce with each factor in the LCD. So we're just gonna be left with seven equals. Now, when we multiply the next fraction, the X minus three will cross reduced with the X minus three in the LCD, leaving us with just two acts on top the numerator times the remainder of the LCD X minus two. And then when we multiply the next fraction X minus two, that is gonna cross reduce with the L C D of X minus two. So we're gonna be left with the numerator X Times X minus three. All right. Now what we're gonna want to do is simplify fully by distributing and seeing what we get, whether it's a linear, quadratic or what type of function we get to solve. Okay, so distributing in, we have two X squared, minus four X and then distributing the negative accent. Here we get negative X squared plus three X and now it looks like we have a quadratic. So I'm gonna simplify everything and move everything. The one side setting one side 20 So two x, my squared minus X squared is going to give this just X squared. Negative for X plus three X is negative one X and I'm gonna minus the seven over, giving us negative seven over on the right hand side. Now we can see if this is factory herbal, but it's not. Nothing multiplies to negative seven, but adds to negative one. So I'm gonna use the quadratic formula. X equals negative. Be so negative. Negative one lost her minus the square root of B squared. So negative one square minus four times a one time C seven negative seven all divided by two a two times one. All right. Simplifying that a little bit more. We get X equals one plus or minus Negative one squared is one and then this is positive. 28. So it's going to be 29. One plus 28 is 29 in the radical, all divided by two. Now we cannot simplify that anymore cause 29 cannot be reduced. The square to 29 cannot be reduced to a, um to a different square root. So this is gonna be our final answer. One plus 29 square to 29 divided by two and one minus the square to 29 divided by two. Now we always have to check to make sure that those answers are acceptable. There's a few things that X cannot equal. In this case, it's whatever is going to make anything in the denominator zero. So three minus three would be zero. So x cannot equal three and cannot equal to. And okay, so this since neither of these answers are positive three or positive to it's okay. The answer is one plus or minus two squared 29 divided by two.

For this question. I'm gonna solve this equation here. What we want to do first is completely clear out our denominators. Now we need to multiply the entire equation by the LCT. Now, the LCD in this case is going to be X plus six square. It needs to be representative of each denominator and have at least the entire expression of each denominator in the LCD. So X plus six squared will represent both of these denominators here. Now we're gonna multiply every single thing by the LCD. All right, so I want to go ahead and multiply that to buy the LCD just X plus X squared. Now, unfortunately, there is no denominator here, too. Reduce anything, So it's just gonna be two times X plus six square, and I would expand that out. So we're gonna be multiplying that out. Most likely. Now multiply this entire fraction by the LCD. Now, thankfully, the X plus six in the Dominator will cancel out with one of the X plus six. It will be left with negative seven the numerator temps just one X plus six year and then equals now, multiplying the second of the 3rd 2nd fraction but the third term here, Bulls X plus six is in the denominator will cancel out the X plus sixes in the L C. DS. That's awesome. Will just be left over 15. The denominator will entirely cross reduce with the LCD. Awesome. Now we can multiply everything out to see what we get. So go ahead and multiply out. This is by no meals. First, that's gonna be X square. This will be plus 12 X plus 36. And then I'm gonna distribute this negative seven b negative 42 equals 15 distributing now the to end on the train. Oh, meal two X Square plus 24 X plus 72 now bringing everything down. And I might as well move this 15 over to the left side. Since it is a quadratic, we want to put it in standard for all right now we can simplify a little bit. There is no other X square terms, so I'm gonna bring that down. 24 x minus seven X is gonna be 17 x positive. 17 x. All right, then we have 70 to minus 42. That is gonna be 30 and then minus 15 is positive 15 equals zero. All right. Dealing with some tricky numbers here. I can now see, since it is a regular, since it is a regular quadratic here, we could see the factor of a Lorries the quadratic formula to solve. But I noticed that this is factor of all. But the quadratic formula may be the best option to use here, but I have to talk to the side a little bit. And I got X Plus one and the other one I got IHS Shula eggs plus 15 equals 20 All right, now that we have both of those factors or reviews the quadratic formula, you consent these equal to zero to solve. So we should get X equals negative one. Yeah. This one will be X equals negative 15 divided by two or negative 7.5. And those there were two final answers.

Weather today, we're going to solve the equation. X squared plus six X minus seven is equal to zero. We're going to find two numbers that add up to get six and multiply to get minus seven. Since, um, the numbers will multiply to get negative number minus seven. We know that one of these must have a minus sign. So let's look at the factors of seven. We have one and seven. In fact, minus one time seven gives us minus seven on seven. Minus one is a bit of six. So we have our two numbers. This tells us that our factory ization is X minus one times x plus seven sequitur zero that from this we know that one of the bracketed terms must equal to zero. So either X minus one is equal to zero. Giving us ex secret one or X plus seven is equal to zero. Giving us X is equal to minus seven. And here are two solutions. Thank you

For this problem, you again distribute the negative sign over the second set of parentheses. So it goes to negative seven X cubed plus two X squared, minus six. And if you look, each like term, cancels the other out so seven X cubed minus seven x Q. Become zero negative two X squared plus two x squared. Become zero and six months. Six equals zero. So the final answer is zero.


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