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CCSS.Math: ,

use the quadratic formula to solve the equation zero is equal to negative seven Q squared plus 2 Q plus 9 now the quadratic formula it applies to any quadratic equation of the form we can put the 0 on the left hand side 0 is equal to a x squared plus BX plus C and we we generally deal with X's in this problem we're dealing with Q's but the quadratic formula says look if you have a quadratic equation of this form that the solutions the solutions of this equation are going to be X is going to be equal to negative B negative B plus or minus the square root of b squared minus 4ac all of that over 2a and this is actually two solutions here because there's one solution where you take the positive square root and there's another solution where you take the negative solution the negative root so it gives you both roots of this so if we look at the the quadratic equation that we need to solve here we can just pattern match we're dealing with Q's not X's but this the same general idea it could be X as if you like and if we look at it negative seven corresponds to a that is our a it's the coefficient on the second degree term two corresponds to B it is the coefficient on the first degree term and then nine corresponds to C it's the constant it's the constant so let's just apply the quadratic form of the quadratic formula will tell us that the solutions the Q's that satisfy this equation Q will be equal to negative B B is two plus or minus plus or minus the square root of B squared of B squared of two squared 2 squared minus minus four times a times negative 7 times negative seven times C times C which is nine times nine and all of that over 2a all of that over all of that over two times a which is once again negative 7 which is negative 7 and then we just have to evaluate this so this is going to be equal to this is going to be equal to negative 2 negative 2 plus or minus the square root of C 2 squared is 4 and then if we just take this part right here if we take the negative 4 times negative 7 times 9 this negative and that negatives going to cancel out so it's just going to become a positive number and 4 times 7 times 9 4 times C 4 times 9 is 36 36 times 7 let's do it up here 36 times 7 7 times 6 is 42 7 times 3 or 3 times 7 is 21 plus 4 is 25 252 so this becomes 4 plus 252 remember this is a you have a negative 7 and you have a minus out front those cancel out that's why we have a positive 252 for that part right there and then our denominator our denominator 2 times negative 7 is negative 14 now what is this equal what is this equal well we have this is equal to negative 2 plus or minus the square root of what's four plus 252 is just 256 all of that over negative 14 and what's 256 that's what's the square root of 256 it's 16 you can try it out for yourself this is 16 times 16 so the square root of 256 is 16 so we can rewrite this whole thing as being equal to negative 2 plus this plus 16 over negative 14 or negative 2 minus right this is plus 16 over negative 14 or minus 16 or minus 16 over negative 14 right if you think of as plus or minus that plus is that plus right there and if you have if you have that minus that minus is that - right there now we just have to evaluate these two numbers just have to evaluate them negative 2 plus 16 is 14 divided by negative 14 is negative 1 so Q could be equal to negative 1 or negative 2 minus 16 is negative 18 divided by negative 14 is equal to 18 over 14 the negatives cancel out which is equal to 9 over 7 so Q could be equal negative 1 or it could be equal to 9 over 7 and you could try these out substitute these Q's back into this original equation and verify for yourself that they satisfy it we could even do it with the first one so if you take Q is equal to negative 1 negative 7 times negative 1 squared negative 1 squared is just 1 so this it would be negative 7 times 1 right that's negative 1 squared negative 1 times 2 is minus 2 plus 9 so it's my it's negative 7 minus 2 which is negative 9 plus 9 does indeed equal 0 so this checks out and I'll leave it up to you to to verify that 9 over 7 also works out