The graph will intersect the x axis when y=0.sparkle6 wrote:In the xy plane, at what points does the graph of y = (x+a)(x+b) intersect the x-axis?
1) a + b = -1
2) The graph intersects the y axis at (0, -6)
[spoiler]Answer: C[/spoiler]
Since y = (x+a)(x+b), y=0 when either (x+a) = 0 or (x+b) = 0.
Thus, to determine which x values will make y=0, we need to know the values of a and b.
Statement 1: a+b = -1
2 variables, 1 linear equation, insufficient.
Statement 2: The graph intersects the y axis at (0, -6)
Substituting x=0 and y=-6 into y = (x+a)(x+b), we get:
-6 = (0+a)(0+b)
-6 = ab.
2 variables, 1 equation, insufficient.
Statements 1 and 2 together:
Substituting b = -a-1 into ab = -6:
a(-a-1) = -6
-a²-a+6 = 0
a²+a-6=0
(a+3)(a-2)=0
a=-3 or a=2.
If a=-3, since ab=-6, b=2.
If a=2, since ab=-6, b=-3.
Each case yields the same combination of values: 2 and -3.
Thus, the equation of the graph is y=(x+2)(x-3), and the graph will cross the x axis at x=-2 and at x=3.
Sufficient.
The correct answer is C.
