The Discriminant
Learn how the discriminant reveals the nature of quadratic solutions before solving.
Definition
| Discriminant | Number of Solutions | Type of Solutions |
|---|---|---|
| Two solutions | Two distinct real numbers | |
| One solution | One repeated real number | |
| No real solutions | Two complex numbers |
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Worked Examples
Determine the nature of solutions for
Identify a, b, and c
, , → Coefficients identified
Calculate the discriminant
→
Interpret the result
Since , there are two distinct real solutions → Two real solutions
Verify by solving
, so or → and confirmed
Answer: Two distinct real solutions: and
Common Mistakes
Forgetting the negative sign when is negative
Why it's wrong: When , students calculate instead of
Correct: Always square the entire coefficient including its sign:
Confusing with
Why it's wrong: Students sometimes forget to square , drastically changing the result
Correct: The discriminant is . The MUST be squared.
Thinking means no solutions
Why it's wrong: Zero seems like nothing, so students assume no solutions exist
Correct: means exactly ONE solution (a repeated root), not zero solutions
Misidentifying coefficients when equation is not in standard form
Why it's wrong: Students use wrong values for , , if equation is like
Correct: Always rewrite in standard form first:
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Practice Problems
15 problemsWhat is the discriminant formula for ?
Why It Matters
- Efficiency: Before spending time solving, check if real solutions even exist
- Graphing: Know whether a parabola crosses the x-axis (and how many times) without graphing
- Problem solving: In word problems, verify that your setup produces valid real-world answers
- Engineering: Determine if a trajectory, circuit, or design has feasible solutions
Real World Applications
Projectile Motion
When launching a projectile, the discriminant tells us whether it will reach a certain height.
Example:
A ball thrown upward follows . To find when : . The discriminant tells us the ball never reaches 25 meters.
A rocket's height is modeled by . You want to know if it reaches 64 meters.
Does the rocket reach 64 meters? Use the discriminant to decide.
Step 1: Write the mathematical expression
Set up , then calculate :
Business Break-Even Analysis
Companies use the discriminant to determine if profit targets are achievable.
Example:
If profit , finding when gives . With , profit of 150 is impossible.
A company's daily profit is thousand euros, where is items sold (in hundreds).
Can they achieve a profit of 16 thousand euros?
Step 1: Write the mathematical expression
Set : . Find :
Key Takeaways
- 1The discriminant is for the equation
- 2If : two distinct real solutions
- 3If : one repeated real solution (double root)
- 4If : no real solutions (two complex solutions)
- 5The discriminant appears under the square root in the quadratic formula
- 6Use the discriminant to predict solutions without fully solving the equation
Frequently Asked Questions
Glossary
- Discriminant
- The expression that determines the nature of solutions to a quadratic equation
- Double root
- A repeated solution that occurs when the discriminant equals zero
- Real solution
- A solution that is a real number (not involving )
- Complex solution
- A solution involving imaginary numbers, occurring when
Formula Card
Discriminant
The expression that determines the nature of solutions
Quadratic Formula
Formula for solving any quadratic equation
Two Solutions
Condition for two distinct real solutions
One Solution
Condition for one repeated real solution
No Real Solutions
Condition for no real solutions