Discriminant and Number of Solutions
What You’ll Learn
Section titled “What You’ll Learn”In this lesson you’ll learn what the discriminant is, how to calculate it, and how it tells you the number and type of solutions a quadratic equation has.
The Concept
Section titled “The Concept”The discriminant is the expression under the square root in the quadratic formula:
It acts as a “preview” of the solutions before you even solve the equation.
The diagram shows three parabolas, one for each case. The green parabola (D > 0) crosses the x-axis twice, giving two real solutions. The gold parabola (D = 0) just touches the x-axis at one point, giving one repeated root. The orange parabola (D < 0) never reaches the x-axis at all, meaning there are no real solutions (only complex/imaginary ones).
What the discriminant tells us:
- D > 0 → two distinct real solutions (parabola crosses x-axis twice)
- D = 0 → one real solution, a repeated root (parabola touches x-axis once)
- D < 0 → no real solutions, two complex/imaginary solutions (parabola doesn’t reach x-axis)
This is very useful because it helps you know what kind of answer to expect and whether you need complex numbers.
Worked Example
Section titled “Worked Example”For each equation, calculate the discriminant and state the number of real solutions.
1. Two real solutions: x² − 6x + 5 = 0
Here a = 1, b = −6, c = 5.
D = 16 > 0, so there are two distinct real solutions.
2. One real solution: 2x² − 8x + 8 = 0
Here a = 2, b = −8, c = 8.
D = 0, so there is exactly one real solution (a repeated root).
3. No real solutions: x² + 4x + 7 = 0
Here a = 1, b = 4, c = 7.
D = −12 < 0, so there are no real solutions, only complex (imaginary) solutions.
Real-World Application
Section titled “Real-World Application”The discriminant helps in practical decision-making:
- Business: will this pricing model give two break-even points, one, or none?
- Physics: will a projectile hit a target twice, once (tangent), or miss completely?
- Engineering: does this quadratic model predict two realistic solutions or impossible ones?
- Finance: analyzing when profit turns positive.
Example: a company’s profit model is p(x) = −2x² + 120x − 800. Calculating the discriminant tells them whether there are two break-even points or if the business model needs adjusting.