In Exercises 101–106, solve each equation.

|x^2 + 2x - 36| = 12

Question

In Exercises 101–106, solve each equation.

|x^2 + 2x - 36| = 12

Accepted Answer

Hello today we're going to be solving the following equation. So the equation given to us is the absolute value of X squared minus seven. X minus four is equal to 14. Well in order to start solving this we need to go ahead and open up the absolute value quantity. In order to do that you have to set it equal to both a positive and negative number. So we're going to end up with two different equations because of the absolute value quantity we're going to have X squared minus seven. X minus four is equal to 14. And we're also going to have a second equation where x squared minus seven, X minus four is equal to negative 14. Now let's go ahead and start with the left hand side. Well in order to solve for X we need to go ahead and set this quantity equal to zero. In order to do that we can subtract 14 to both sides of the equation. Doing so is going to give us X squared minus seven. X minus 18 is equal to zero. And this trin Amiel is factory herbal, It factors to the quantities X minus nine times X plus two is equal to zero. And here we can go ahead and apply our zero theorem and set both of the quantities equal to zero. So we get x minus nine is equal to zero and X plus two is equal to zero, solving both of these is going to give us X is equal to nine and x is equal to negative two. So we have two values of X. Now let's go ahead and solve the right hand side. So once again we want to go ahead and set this equation equal to zero in order to sell for X in order to do that, I'm gonna go ahead and add 14 to both sides of this equation. Doing so is going to give me X squared minus seven, X plus 10 is equal to zero and this try no meal is factory able and if factors two, X minus five times x minus two equal to zero. And once again we can use our zero property to set both of these quantities equal to zero. So we get X minus five is equal to zero and x minus two is equal to zero. And solving for both quantities is going to give me X is equal to five and X is equal to two. So that's two more quantities X that we get. So in total we have four possible solutions for this equation, we have X is equal to nine, X is equal to negative two, X is equal to five and X is equal to positive two. But now we need to go ahead and check to see if these values of X work with our equation. So let's go ahead and do that. So when X is equal to nine, we get the absolute value of nine squared minus seven times nine minus four is equal to 14, while nine squared is going to be 81 7 times nine is going to be 63 so nine So simplifying this is going to give us 81 minus -4 is equal to 14. And simplifying the absolute value is going to give us the absolute value of is equal to 14. Now again the absolute value lets us represent any number as a positive number, so we are left with 14 is equal to 14 and that's a true statement. So X is equal to nine is one of our solutions. Now we're gonna go ahead and repeat this process for the other three numbers. So when X is equal to negative two we get the absolute value of negative two squared minus seven times negative two minus four is equal to 14, negative two squared is going to give us four and seven times negative two is gonna give us negative 14 and negative times negative gives us a positive So we are left with the absolute value of four plus minus four is equal to 14 and that's going to simplify to the absolute value of 14 equal to 14 and this gives us the statement 14 is equal to 14 which is true, So that means that X is equal to -2 is another solution. Now we're gonna go ahead and check X is equal to five, plugging this into our equation is going to give us the absolute value of five squared minus seven times five minus four is equal to 14. 5 squared is going to give us 25 7 times five is going to give us 35. So we have the absolute value of 25 minus 35 minus four is equal to 14, simplifying. This is going to give us the absolute value of negative Is equal to 14 and we can go ahead and represent -14 as positive 14 because of the absolute value. So we get 14 Is equal to 14 which is a true statement. So that means that X is equal to five is another solution. And finally checking X is equal to two. We get the absolute value of two squared minus seven times two minus four is equal to 14, 2 squared is going to give us four and seven times two is going to give us 14. So we have the absolute value of four minus 14 minus four is equal to 14 and this is going to simplify to negative 14 is equal to 14 and the absolute value allows us to represent negative 14 as a positive number. So we are left with 14 is equal to 14 which is a true statement and that means that X is equal to two Is one of our solutions. So we have just verified that the four numbers that we solved for earlier are all possible solutions of this equation and that means that the solution to this problem is going to be a So I hope this video helps me understanding how to solve an absolute value equation, and I'll go ahead and see you all in the next video.

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