Quadratics & Parabolas (Advanced)

Harder items at competition-adjacent depth.

Phase 2 · SAT Math

Quadratics & Parabolas: core idea

Completing the square for vertex.

Solve $x^2+4x+1=0$ .

Hint: Use the quadratic formula. Answer: -2±√3.

1. For $y=-x^2+10x-21$ , find the maximum value.
2. If $r$ and $s$ are roots of $x^2-6x+1=0$ , find $r^2+s^2$ .
3. The graph of $y=ax^2+bx+4$ passes through $(2,0)$ and $(4,0)$ . Find $a$ .
4. Solve $\frac{x^2-9}{x-3}=8$ , with $x\ne3$ .
5. A projectile's height is $-16t^2+64t+80$ . When does it hit the ground?
6. Find $k$ so that $x^2+kx+36=0$ has roots in ratio $1:4$ and both roots are negative.
7. If $x+\frac1x=5$ , find $x^2+\frac1{x^2}$ .
8. The quadratic $x^2-2px+p^2-9=0$ has roots differing by $6$ . Find $p$ .
9. A parabola has vertex $(1,-4)$ and passes through $(3,4)$ . Write its equation.