Describe the solutions to the following quadratic equation:

`
expr(["+",
["*", A, ["^", "x", 2]],
["*", B, "x"],
C])
` = 0

- One rational solution
- Two rational solutions
- Two irrational solutions
- One complex solution
- Two complex solutions

We could use the quadratic formula to solve for the solutions and see what they are, but there's a shortcut...

```
\qquad
x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}
```

$("#quadratic").text("\\qquad x = \\dfrac{-b \\pm" +
" \\sqrt{\\blue{b^2 - 4ac}}}{2a}");
MathJax.Hub.Queue(["Reprocess", MathJax.Hub,
$("#quadratic")[0]]);

Think about what the part of the quadratic
formula under the
radical tells us about the solutions.

Substitute the `a`

, `b`

, and
`c`

coefficients from the quadratic
equation:

```
\qquad\begin{array}
&& b^2-4ac \\ \\
=&
```

`B`^2 - 4 (
`A`)(`C`) \\ \\
=& `DISCRIMINANT`
\end{array}

Because `\blue{b^2 - 4ac} = 0`

, then the
quadratic formula reduces to
`\dfrac{-b}{2a}`

, which means there
is just one rational solution.

Because `\blue{b^2 - 4ac}`

is negative, its
square is imaginary and the quadratic formula reduces to
`\dfrac{-b \pm \sqrt{`

, which means there are two complex solutions.
`DISCRIMINANT`}}{2a}

Because `\blue{b^2 - 4ac}`

is a perfect
square, its square root is rational and the
quadratic formula reduces to
`\dfrac{-b \pm `

, which means there are two rational solutions.
`sqrt(DISCRIMINANT)`}{2a}

Because `\blue{b^2 - 4ac}`

is not a perfect
square, its square root is irrational and the
quadratic formula reduces to
`\dfrac{-b \pm \sqrt{`

, which means there are two irrational solutions.
`DISCRIMINANT`}}{2a}