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How many real solutions does a quadratic equation have when the discriminant is positive, negative, and zero?
| Equation | Discriminant |
|---|---|
| f(x)=0 | Negative |
| g(x)=0 | Positive |
| h(x)=0 | Zero |
| j(x)=0 | Zero |
The information whether the discriminant is positive, negative, or zero can be used to determine the number of real solutions of a quadratic equation. However, it works both ways — the number of real solutions tells us whether the discriminant is positive, negative, or zero.
| Discriminant | Number of Solutions |
|---|---|
| Positive | Two real solutions |
| Negative | No real solutions |
| Zero | One real solution |
We can see that f has no x-intercepts, g has two x-intercepts, and h and j have one x-intercept each. Therefore, f(x)=0 has no real solutions, g(x)=0 has two real solutions, and h(x)=0 and j(x)=0 have one real solution each. Let's make a table to see what it means in terms of the dicriminants!
| Equation | Number of Solutions | Discriminant |
|---|---|---|
| f(x)=0 | No real solutions | Negative |
| g(x)=0 | Two real solutions | Positive |
| h(x)=0 | One real solution | Zero |
| j(x)=0 | One real solution | Zero |