Quadratic Equations and Graphs

Remember the following.

Given  ax² + bx + c = 0
sum of roots = -b/a
product of roots = c/a
a > 0 = happy = open upwards
a < 0 = unhappy = open downwards
c = y-intercept
-b/2a = x-coordinate of the vertex

Answer : B

Note that (2^{p – 2})(2^{q – 2}) = 2^{p + q - 4}
 (Remember we sum the power together, not multiply)

Besides, p + q = –(–1) / 1 = 1

Hence, the answer is 2^-3 = 1/8

Answer : E

By observation, everyone knows x can be a.
For the remaining solution, you may just substitute x = 2,3 and 5 – a to see which one is correct.
Of course, you may expand it and solve the equation.

Answer : E

Let y1 = ax² + bx + c
Let y2 = mx + k

For α ≤ x ≤ β, we can see that y2 ≤ y1. Therefore, y1 – y2 ≥ 0.

Answer : E

Since it has real roots, discriminant is greater than or equal to 0.
When we arrive at the step k(k - 4) ≥ 0, we can draw the graph y = x(x - 4) to help us.

Answer : B

Since it smiles, a > 0.
Note that the y-intercept is greater than 0. Thus, c > 0.
Note that the curve doesn't touch the x-axis. It means y = 0 has no solution and the discriminant is smaller than 0. Thus, b² - 4ac < 0

Answer : D

Given y = a(x – h)² + k
Vertex is (h,k)
Axis of symmetry is x = h 

For A, the vertex is (-1, -4)
For B, the axis of symmetry is x = -1