Algebra 2 / Unit 2: Quadratic Functions and Complex Numbers
2.6 Quadratic Inequalities
Notes
To solve $ax^2 + bx + c < 0$ (or $>$, $\le$, $\ge$):
- Find the zeros of the quadratic. They split the number line into intervals.
- Test a number in each interval, or think about the graph: when $a > 0$, the parabola is below the axis between the zeros and above it outside them.
- Example: $x^2 - 4 > 0$ has zeros $\pm 2$, so $x < -2$ or $x > 2$.
- Use $\le$ or $\ge$ to include the zeros.
Practice
Solve $x^2 - x - 6 < 0$.
Zeros are $3$ and $-2$. The parabola opens up, so it is negative between them.
Solve $x^2 - 9 \ge 0$.
Zeros are $\pm 3$; the parabola is at or above the axis outside them.
A ball's height is $h(t) = -16t^2 + 64t$ feet after $t$ seconds. When is it higher than 48 feet?
$-16t^2 + 64t - 48 > 0 \Rightarrow t^2 - 4t + 3 < 0 \Rightarrow (t - 1)(t - 3) < 0$.
Ready for more? Take the Unit 2 test.