y = (x + a)(x + b)
y = x^2 + ax + bx + ab
y = x^2 + (a+b)x + ab
The prompt tells us that it crosses y-axis at (0,-6), therefore:
-6 = 0^2 + 0 + ab
-6 = ab
And, to figure out when this equation crosses the x-axis, we set y = 0, and solve the equation:
0 = (x+a)(x+b)
x = -a or -b
So in order to figure out what a and b are equal to, I need to find two numbers that add up to 1 (since a + b = 1) and multiply together to get -6 (since ab = -6). I can do this in my head. I hope you can do.
(This is my mind thinking...)
Since they multiply together to get -6, I know that one is positive and and one is negative.
Since they add up to 1 and one is negative and one is positive, I know that their respective absolute values have a difference of 1. And since, they add up to 1 and not -1, I know that the absolute value of the positive number is bigger than the absolute value of the negative number.
So, what two numbers multiply together to get 6? And have a difference of 1?
3 and 2!
So 3 and 2 are the absolute values of a and b.
And since they need to add up to 1, then a and b are 3 and -2
Thus, x = -3 or 2 (since x = -a or -b)
The intersection points are (-3,0) and (2,0).
