  In the system of equations above, , , and are constants. For which of the following values of , , and does the system have no solutions?

A)
B)
C)
D) John Lockman says:

lucky guess y= a( x-h)^2+k h and k are the order pair (x,y) respectively. if A is negative, then the parabola opens down so therefore k < than the Y solution which is straight line for all xs. the h value is not important just the sign of a must be negative and if so then finding a solution k ( or here c) that is < Y;An up turned flashlight will shine on the ceiling but as long as the flashlight is shining downward and lower then the ceiling ( 5 units high) it will not shine on it. aarhin10 says:

This question doesn’t even require you do to any actually calculations! Like the explanation said, it’s helpful to know that this is in vertex form, so (b,c) is your vertex.

If y = 5, and the question is looking for the answer that will provide NO solutions, that means that the line (which is in vertex form) cannot cross y = 5 (visualizing this helped me out big time…you could sketch a simple graph)

So:
– With parabolas, if it’s positive, a > 0
– With parabolas, if it’s negative, a < 0
– Our vertex (b,c) must either be below 5 or above 5…(when plotted)

Our a value is negative, so the maximum (vertex) must be below y = 5. We see that the vertex is (1,7), and this surpasses y = 5 and would cause 2 intersections to occur. So, answer choice A is wrong.

Our a value is positive, so the minimum (vertex) has to be above y = 5. We see that the vertex for B is (1,-3) and for C, (-3, 2). Both of these vertex(s) fall below y = 5 and would cause intersections to occur. So, answer choice(s) B and C are both wrong as well.

Now let’s check Answer choice D.
Our a value is negative, so the maximum (vertex) has to be below y = 5. We see that the vertex is (5, 4)…the y value, 4, is lower than the y value, 5, so, answer choice D is correct.

Final answer: a = -5, b = 5, c = 4 infoanalysis says:

that vertex formula is important to remember y =a(x-H)^2+K where (H,K) are vertexes I KEEP FORGETTING THE FORMULA BUT REMEBER THAT IT IS THE KEY HERE.
THEN you got to remember that parabolas with a ” -a” are open down ( negative makes a sad smile or Micky d arches are negative or ” it’s positive to open up.” years of therapy drilled that in my head until I met math)
after that its simple(“Mrs Lincoln”)

I have to remember that the constant will always represent the maximum for neg (open down) and minimum (for pos open up) on the Y value because it hangs out no matter what x is and then that which you subtract from the x is the place holder of that Y value on the parabola.