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one of the roots is 3 that means x=3 or , adding -3 to both sides, x-3=0 let the other root be called "a" then we know (x-3)(x-a)=0 multiply that out x^2 -3x-ax +3a = 0 or x^2 -(3+a) x + 3a =0
match up x^2 -(3+a) x + 3a =0 x^2+Kx−6=0 we see 3a= -6 solve for a by dividing both sides by 3 we get a= -2
using a= -2 in x^2 -(3+a) x + 3a =0 x^2 -(3-2) x +3*-2 = 0 simplifying: x^2 - (1) x -6 =0 or x^2 -x -6 =0
it looks like K= -1
why is one of the roots 3?
It says One of the roots of the equation x^2+Kx−6=0 is 3
Hey how do i know if it's (x-3)(x-a) and not (x+a) ?????????
I tried (x-3)(+a) and it worked out better
say you factored a quadratic and got (x-3)(x-2) = 0 the roots are the x values that make this expression zero. this will be zero if x=3 (which makes 3-3 zero) or x=2 (which makes 2-2 zero) if you had (x-3) (x+2)=0 the second root is when x= -2 notice if we write (x+2) as (x - (-2)) the number -2 is the root. in other words, if we write (x-a) then a will be the root (the x value that makes the expression zero)
btw, what is question ? It looks like it was cut off at the top.
It's asking which one is greater a or b
so in what cases would it be (x-3) (x+a)
in this case , K = -1, so they are equal
I'm still having a hard time understanding how I would know for sure that it would be (-)(-) instead of (-)(+) or (+)(-)
Let's factor x^2 -x -6 =0 the -6 means 1) different signs on the roots, 2) they multiply to give us 6 1,6 2,3 are the choices the -1x means the larger nuumber is negative, and they add to -1 so 2 and -3 are the factors (x+2)(x-3)=0 or (x - (-2)) (x-3) = 0
I guess the point is you can always write the factors as (x - a) where a is the root. if the root were -2 x= -2 makes the expression zero, then the factor would be (x- -2) (which is the same as (x+2) )
But I think I am being a clear as mud here.