To Be Determined

Learning Focus

Use the discriminant to determine the number and type of solutions.

Big idea

For $ax^2+bx+c=0$, the discriminant is $D=b^2-4ac$.

• $D>0$: two real solutions (rational if $D$ is a perfect square).
• $D=0$: one real (repeated) solution.
• $D<0$: no real solutions (two complex).

Worked example

$x^2+4x+5=0$: $D=16-20=-4<0$. No real solutions; the parabola never touches the $x$-axis.

Takeaways

• $D$ is the expression under the square root in the quadratic formula.
• $D$ predicts the number of $x$-intercepts.
• $D=0$ means the vertex lies on the $x$-axis.