The Discriminant
Use the discriminant when the question asks how many real roots exist.
Where it comes from
In , the term under the square root decides how many real answers exist. It is the .
The three cases
Distinct roots are different roots. Equal roots are the same value repeated. A positive number under the square root gives two choices from ; zero makes those choices equal; a negative number has no real square root.
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| Discriminant | The equation has | The curve |
|---|---|---|
| two distinct real roots | cuts the -axis at two separate points | |
| two equal real roots | touches the -axis at exactly one point | |
| no real roots | misses the -axis completely |
- Discriminant
- The equation has
- two distinct real roots
- The curve
- cuts the -axis at two separate points
- Discriminant
- The equation has
- two equal real roots
- The curve
- touches the -axis at exactly one point
- Discriminant
- The equation has
- no real roots
- The curve
- misses the -axis completely
For y = x² + c: b² − 4ac = 12.0, so there are two different real roots.
Finding a constant from the number of roots
A parameter such as is a constant whose allowed value the question asks you to find. Treat it as fixed while solving in .
Find the set of values of for which has no real roots.
Show worked answer
Worked example
Find the values of for which has two equal roots
- Rearrange to zero.
- Equal roots means .
Common mistake
Finding a range of values
Worked example
Find the values of for which has two distinct roots
- Two distinct roots means . Also require so the equation remains quadratic.
- The quadratic in is positive outside its roots.
Find the set of values of the constant for which the quadratic equation has two distinct real roots.
Show worked answer
This is the published mark-scheme answer. Strictly, removes the term and makes the equation linear, so its highest power of is now. Keep that degeneracy check in a new problem, but do not change the published answer for this paper.