Understand the method
The quadratic formula contains \(\sqrt{b^2-4ac}\). Over the real numbers, a positive quantity has two opposite square roots, zero has one, and a negative quantity has none.
| Discriminant | Distinct real roots | Graph |
|---|---|---|
| \(\Delta>0\) | Two | Crosses the x-axis twice |
| \(\Delta=0\) | One repeated root | Touches the x-axis |
| \(\Delta<0\) | None | Does not meet the x-axis |
For example, \(x^2+1=0\) has \(\Delta=-4\), so it has no real roots. This does not mean there are no complex roots.
Three worked examples
Example 1
Count the roots of \(2x^2+3x-2=0\)
Use \(a=2,b=3,c=-2\).
- \[\Delta=3^2-4(2)(-2)=25\]
The discriminant is positive, so there are two distinct real roots. You do not need to calculate the roots to answer this question.
Keep the negative sign of c inside brackets.
Example 2
Count the roots of \(x^2-4x+4=0\)
- \[\Delta=(-4)^2-4(1)(4)=0\]
There is one repeated real root. Indeed, \((x-2)^2=0\) gives \(x=2\).
A repeated root is one distinct value, counted twice by multiplicity.
Example 3
When does \(kx^2+4x+1=0\) have two distinct real roots?
First require \(k\ne0\) so that the equation is quadratic.
- \[\Delta=16-4k>0\]
Thus \(k<4\), with \(k\ne0\). When \(k=0\), the equation is linear and has just one root.
A discriminant inequality alone is not enough if the leading coefficient can vanish.
Apply and explain
Connect algebra to intersections
Equating a line and a quadratic curve gives a quadratic equation. Its discriminant tells you how many intersection points there are, provided the substituted equation remains quadratic.
Exam-style check: y = x² + 1 and y = 2x
The discriminant is zero, so there is one intersection: \((1,2)\). Show the rearrangement, discriminant and matching coordinates for a complete argument.
These study examples explain the reasoning; they are not an official mark scheme.
Common mistakes to catch
- Using the wrong sign for a coefficient
- In \(x^2-6x+7\), substitute \(b=-6\), including brackets when squaring.
- Saying two distinct roots when the discriminant is zero
- Zero gives a repeated root, with one distinct real value.
- Forgetting the quadratic condition
- Exclude parameter values that make \(a=0\), and consider that case separately.
Now try it yourself
Put the method into practice
Return later and classify \(3x^2+2x+1=0\) without solving for x. Explain what the graph does at the x-axis.