In a xy plane, at what two points, does the graph y = (x+a)(x+b) intersects the x axis.
i) a+b = -1
ii) the graph intersects y axis at point (0,-6)
i) a+b = -1
ii) the graph intersects y axis at point (0,-6)
First, we know that whenever a graph crosses the x axis, y=0. So, in essence, we want to know the values for x when 0 = (x+a)(x+b). Anytime we multiply two values to get a product of 0, one (or both) of the elements must equal 0. So (x+a) = 0, which means x = -a; or (x+b) = 0, which means x = -b. (Or both.) In other words, if we have a and b, we have our values of x.
1) clearly insufficient. You could have a = -1 and b = 0 or a = -1/2 and b = -1/2
2) clearly insufficient. This gives us the inflection point of the parabola, (the bottom of the smile) but it doesn't tell us where we cross the x axis.
Or consider the algebra. At this point x = 0 and y = -6. Substituting into the original equation, we'd get -6 = ab. Maybe a =2 and b = -3 or a = -1 and b = 6.
T) Sufficient.
Now we have a+ b = -1 from S1 and -6 = ab from S2. So let's do some algebra:
a = -1 - b. Plug into the second statement to get -6 = (-1 -b)b
-6 = -b - b^2
b^2 + b - 6 = 0
(b +3)(b -2) = 0
b = -3 or b = 2
So either b = -3 and a = 2 or b = 2 and a = -3. Either way, we'll have our two values for x.
