now sometimes you may encounter difficult trinomials trinomials that are just very hard to factor consider this one how would you factor 12x squared plus 243x minus 189 how would you do it we know that we need to multiply the first or the leading coefficient by the constant term 12 times 189 is negative 2 268. we need to find two numbers that multiply to negative 2268 but add to 243 and as you can see it's going to take a long time to do that fortunately there is another easy way that we can use to factor difficult trinomials and that is the quadratic equation let me show you so we know that b is 243 b squared that's 243 squared which is 59 and 49 minus 4 times a a is 12 c is negative 189 so this is a b and c all divided by 2a or 2 times 12 which is 24. so we now have is negative 243 plus or minus square root 59049 and then let's multiply negative 4 times 12 that's negative 48 and then negative 48 times the negative 189 that's going to be positive 9072 all divided by 24. seventy two plus fifty nine thousand forty nine that's sixty eight thousand one twenty one if we take the square root of that number we're going to get two hundred and sixty one so now what we need to do is separate that fraction into two other fractions so this is negative 243 plus 261 over 24 and it's also negative 243 minus 261 over 24. there's two separate answers now negative 243 plus 261 that's eighteen so we have eighteen over twenty four and we can reduce eighteen over twenty four we can divide both numbers by six so this is going to be 3 over 4. negative 243 minus 261 that's negative 504. both numbers are divisible by 4. 504 divided by 4 is negative 126 24 divided by 4 is 6. now 126 is divisible by 6. it turns out that it's 21. so but it's going to be negative 21. so right now what we have is that x is equal to positive 3 over 4 and x is equal to negative 21. on the equation and the right i'm going to add 21 to both sides if i do that i'll get this x plus 21 is equal to zero this is one of the factors now for the other one i'm going to move the three-fourths to the left side so x minus three over four is equal to zero now i'm gonna multiply both sides by four so four x minus three is equal to zero so if we put it together it's going to be 4x minus 3 times x plus 21. but notice that 4x times x is not 12x squared but if we add a 3 to it it does become 12x squared so this is the factored equation right here and let's prove it let's distribute 3 first 3 times 4x is 12x 3 times negative 9 i mean three times negative three that's negative nine and now let's foil twelve x minus nine and x plus twenty one so twelve x times x that's just going to be 12x squared next we have 12x times 21 and that's positive 252x negative 9 times x is negative 9x and negative 9 times 21 that's negative 189 so if we add the two middle terms together 252 minus nine that's going to give us the 243. so as we can see this is indeed the factored expression so that's how you can use the quadratic formula to basically reverse factor the problem if you ever have a difficult trinomial that you need to factor