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# Solving quadratics by factoring: leading coefficient ≠ 1

## Video transcript

so we have 6x squared minus 120 X plus 600 equals 0 like always pause this video and see if you can solve for X if you could find the x values that satisfy this equation all right let's work through this together so this the numbers here don't seem like outlandish numbers they seem like something that I might be able to deal with and I might be able to factor so let's let's try to do that so the first thing I like to do is see if I can get a a coefficient of 1 on the second degree term on the x squared term and it looks like actually all of these terms are divisible by 6 so if we divide both sides of this equation by 6 I'm still going to have nice integer coefficients so let's do that let's divide both sides by 6 so if we divide the left side by 6 divided by 6 divided by 6 divided by 6 and I divide the right but side by 6 so if I do that and clearly if I if I do the same thing to both sides of the equation then the Equality still holds on the left-hand side I am going to be left with x squared and then negative 1 20 divided by 6 that is C 1 20 divided by 6 is 20 so that's minus 20 X and then 600 divided by 6 is 100 so plus 100 is equal to zero divided by 6 is equal to zero so let's see if we can factor if we can if we can express this quadratic as a product of two expressions and the way we think about this and we've done it multiple times if we have something if we have X plus a times X plus B and this is hopefully review for you if you multiply that out that is going to be equal to that equals 2x squared plus a plus B X plus a B and so what we want to do is see if we can factor this into an X plus a and an X plus B and so a plus B needs to be equal to negative 20 that needs to be a plus B and then a times B right over here that needs to be equal to the constant term needs to be a times be right over there so can we think of two numbers that if we take their product we get positive 100 and if we take their sum we get negative 20 well since their product is positive we know that they have the same sign so they're both going to have the same sign so they're either both going to be positive or they're both going to be negative since we know that we have a positive product and since their sum is negative well they both must both be negative although you can't add up to positive numbers and get a negative so they most they both must be negative so let's think about it a little bit what negative numbers when I add to each to the when I add them together I get negative 20 when I multiply again I get 100 well you could try to factor a 100 you could say well negative 2 times negative 50 or negative 4 times negative 25 but the one that might jump out at you is this is negative 10 times I'll write it this way negative 10 times negative 10 and this is negative 10 plus negative 10 so in that case both our a and our B would be negative 10 and so we can rewrite the left side of this equation as I can rewrite it as X and I'll write it this way it first X plus negative 10 x times X plus negative 10 again X plus negative 10 and that is going to be equal to zero so all I've done is I factored this quadratic or another way these are both the same thing as X minus 10 I could rewrite this as X minus 10 squared is equal to 0 and so the only way that the left-hand side is going to be equal to 0 is if X minus 10 is equal to 0 you could think of this as taking the square root of both sides and doesn't matter if taking the positive or negative square root or both of them it's the square root of 0 0 and so we would say that X minus 10 needs to be equal to 0 and so X adding 10 to both sides of this you have X is equal to 10 is the solution to this quadratic equation up here