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CCSS.Math: ,

use the quadratic formula to solve the equation negative x squared plus 8x is equal to 1 now in order to really use the quadratic equation or to figure out what our A's B's and C's are we have to have our equation in the form ax squared plus BX plus C is equal to 0 and then if we know our A's B's and C's will say that the solutions to this equation are X is equal to negative B plus or minus the square root of b squared minus 4ac all of that over 2a so the first thing we have to do for this equation right here is to put it in this form and on one side of the equation we have a negative x squared plus 8x so that looks like the first two terms but our constants on the other side so let's get the constant out of the left-hand side and get a 0 here on the right-hand side so let's subtract let's subtract 1 from both sides of this equation the left-hand side of the equation will become negative x squared plus 8x minus 1 and then the right-hand side 1 minus 1 is 0 now we have it in that form we have ax squared a is negative 1 so let me write this down a is equal to negative 1 a is right here a is equal to negative 1 it's implicit there you could put a 1 here if you like a negative 1 negative x squared is the same thing as negative 1 x squared B is equal to 8 so B is equal to 8 that's the 8 right there and C is equal to negative 1 and C is equal to negative 1 that's the negative 1 right there so now we can just apply the quadratic formula the solutions to this equation are X is equal to negative B so negative B negative B plus or minus plus or minus the square root of B squared of 8 squared 8 squared minus 4ac - let me do that in that in color - for the greens the part of the formula the colored parts are the things that we're substituting into the formula minus four times a which is negative one times negative one times C which is also negative one times negative one and then all of that let me extend the square root sign a little bit further all of that is going to be over two times a in this case a is negative one so let's simplify this so this becomes negative eight this is negative eight plus or minus the square root of 8 squared is 64 and then you have a negative 1 times a negative 1 these just cancel out just to be a 1 so it's 64 minus 4 that's just that 4 over there all of that over negative 2 so this is equal to negative 8 plus or minus the square root of 60 all of that over negative 2 now let's see if we can simplify the the expression the radical expression here the square root of 60 let's see 60 is equal to 2 times 30 30 is equal to 2 times 15 and then 15 is 3 times 5 so we do have a perfect square here we do have a 2 times 2 in there it is 2 times 2 times 15 or 4 times 15 so we could write the square root of 60 is equal to the square root of 4 times the square root of 15 right square root of 4 times the square root of 15 that's what 60 is 4 times 15 and so this is equal to square root of 4 is 2 times the square root of 15 so we can rewrite this expression right here as being equal to negative 8 plus or minus plus or minus 2 times the square root 2 times the square root of 15 all of that over negative 2 now we can divide both of both of these terms right here are divisible by either 2 or negative 2 so let's divide it so we have negative 8 divided by negative 2 which is positive 4 so let me write it over here negative 8 divided by negative 2 is positive 4 and then you have this weird thing plus or minus 2 divided by negative 2 and really what we have here is two expressions but if we're plus two and we divide by negative two it will be negative one and if we take negative two and divide by negative two we're going to have positive one so instead of plus or minus you could imagine it it's going to be you could imagine it now as being minus or plus but it's really the same thing right it's really now minus or plus if it was plus it's not going to be a - now it's so - it's not going to be a plus minus or plus two times the square root two times the square root of fifteen or another way to view it is that the two solutions here are four minus two roots of 15 and four plus two roots of 15 these are both values of x that will satisfy this equation and if this confuse you what I did turning in plus or minus into minus plus let me just take a little bit of a side there I could write this expression up here I could write this expression as two expressions that's what the plus or minus really is there's a negative eight plus the square let me if there's a negative eight plus two roots of 15 over negative two and then there's a negative eight minus two roots 15 over negative two this one simplifies to negative 8 divided by negative 2 is 4 2 divided by negative 2 is negative 1/2 times this 4 minus square root of 15 and then over here you have negative 8 divided by negative 2 which is 4 and then negative 2 divided by negative 2 which is plus the square root of 15 and I just realize I made a mistake up here when we're dividing it 2 divided by negative 2 we don't have this 2 over here this is just a plus or minus the root of 15 we just saw that when I did it out here so this is minus the square root of 15 and this is plus the square root of 15 so the two solutions for this equation it's good that I took that little hiatus there that little aside there the two solutions X could be 4 minus the square root of 15 or X or and X could be 4 plus the square root of 15 either of those values of X will satisfy will satisfy this original quadratic equation