Use the method described in Exercises 83–86, if applicable, and p...

Question

Use the method described in Exercises 83–86, if applicable, and properties of
absolute value to solve each equation or inequality. (Hint: Exercises 99 and 100 can
be solved by inspection.) | 3x^2 - 14x | = 5

Accepted Answer

Hello everyone. We're going to solve the equation. The absolute value of seven X squared minus 19 X equals 12. First thing to recall is that when we're working with absolute values we need to set up two separate equations. So we're going to have seven X squared minus 19. X equals 12. And we also need to solve seven X squared minus 19 X equals negative 12. Because our absolute value it could have been a 12. It could have been a negative 12. So we're going to start by solving these like we would any other quadratic. So I want to set it equal to zero. So seven X squared minus 19 X equals 12. I'm gonna subtract 12 from both sides. This gives me seven X squared minus 19 X minus 12 equals zero. Now looking at that quickly I'm trying to decide if that is factory ble unfortunately I don't think it is. So we're gonna go back to our quadratic formula Again recall the ace X squared plus bx plus C. Is our general form when we're working with this and that the quadratic formula is that X equals negative B plus or minus the square root of B squared minus four A. C. All over two A. So we're going to use that to solve this equation. So we want to find out what X is and let's see our B is negative 19. So negative negative 19 is positive 19 plus or minus the square root of - squared -4 times a. Which is seven Time C. Which is -12. All of it over two times a. So two times 7. All right there's a lot of numbers going on in there. But we want to simplify what we can So the 19 stays out front plus or minus and then under my radical I need to find out what 19 squared is and 19 squared is 361 - times seven times negative 12. And that would be a positive 336. All of it over two times 7 which is 14. So now adding what's under my radical I have 19 on the outside still plus or minus the square root of C. 697 over 14. And I think that's all I can do with that. That is probably my X. Value for that side. It doesn't look like it simplifies any further. So those are technically two X values because we have a plus and a minus. So we now need to go solve the other equation. The seven X squared minus 19 X equals negative 12. So just like I did before I want to make this equal to zero. So I do that by adding 12 to both sides. So I have seven x squared minus 19 X plus 12 equals zero. Again I want to see if I can factor this first and I actually can in this case if I wasn't sure I could still do the entire quadratic formula but I don't want to have to if I don't have to. So when I factor this I get seven x - times X -1. Because my quick mental check that will give me the middle term of negative 19 X. And multiply to a positive 12. So it is factory ble Alright, so now I'm gonna solve that by setting each set of parentheses equal to zero. So seven X -12 equals zero. An x minus one equals zero, Solving seven X -12. I'm adding 12 to both sides. So I get seven x equals 12, Dividing both sides by seven. I get the X is 12/7 and then for x minus one equals zero. I add one to both sides. I get X equals one. And now I have all of my X values. I have X equals one. X equals 12 7th and X equals 19 plus or minus the square root of 697 over 14. All right. So now to see which answer that matches up with and it matches to answer choice, a answer choice A says that X is one, X is 7 12 and X is 19 plus or minus the square root of 697. Over 14. Have a nice day

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Use the method described in Exercises 83–86, if applicable, and properties of absolute value to solve each equation or inequality. (Hint: Exercises 99 and 100 can be solved by inspection.) | 3x^2 - 14x | = 5