Quadratic Equations — CBSE Class 10 Discriminant Explained Clearly
The Discriminant (b² − 4ac) Predicts the Nature of Roots Before Solving
By Dhruv Chandravanshi | Class 10 Mathematics | CBSE
Have You Ever Wondered How a Mathematician Knows Whether an Equation Has Answers Before Actually Solving It?
In this article, you will learn:
- What a quadratic equation actually is before the algebra appears
- What the discriminant does and why it is a prediction tool
- What b² − 4ac tells us about roots
- What CBSE Class 10 expects in discriminant questions
By the end, you will understand that the discriminant is not simply a step inside the quadratic formula.
It answers a different question entirely:
Not what the roots are, but what kind of roots exist.
A Garden Planning Problem
Arjun is helping his father plan a rectangular vegetable garden behind their house in Patna.
His father wants:
- Area = 24 square metres
- Length = 5 metres more than the width
Arjun writes:
Let width = x metres
Then:
Length = x + 5
Area = length × width
So:
x(x + 5) = 24
Expanding:
x² + 5x = 24
Bringing all terms to one side:
x² + 5x − 24 = 0
Arjun pauses.
This equation contains x².
It is not a simple linear equation anymore.
Before solving it, he wants to know:
Does this equation actually have a real answer?
If not, the garden cannot exist physically.
Before solving, mathematicians often check one expression first.
That expression is called the discriminant.
Why Quadratic Equations Are Different
Linear equations always have one solution.
Example:
2x + 3 = 7
Quadratic equations behave differently.
Their graph forms a parabola, a U-shaped curve.
This curve can interact with the x-axis in three ways:
- It cuts the axis at two points → two distinct real roots
- It touches the axis at one point → equal real roots
- It never touches the axis → no real roots
So mathematicians first ask:
Which of these three situations applies?
The answer comes from the expression inside the square root in the quadratic formula.
The Discriminant
The quadratic formula is:
x = (−b ± √(b² − 4ac)) ÷ 2a
The expression inside the square root is:
D = b² − 4ac
This expression is called the discriminant.
It reveals the nature of the roots before solving the equation.
How the Discriminant Works
Step 1: Write the Equation in Standard Form
Every quadratic equation must be written as:
ax² + bx + c = 0
Step 2: Identify a, b, and c
- a = coefficient of x²
- b = coefficient of x
- c = constant term
Step 3: Calculate the Discriminant
D = b² − 4ac
Step 4: Interpret the Value of D
If D > 0
- Two distinct real roots exist
- The parabola crosses the x-axis at two points
If D = 0
- Two equal real roots exist
- The parabola touches the x-axis once
Root:
x = −b ÷ 2a
If D < 0
- No real roots exist
- The parabola does not intersect the x-axis
Applying the Discriminant to Arjun’s Garden
Equation:
x² + 5x − 24 = 0
Identify coefficients:
- a = 1
- b = 5
- c = −24
Calculate D:
D = 5² − 4 × 1 × (−24)
D = 25 + 96
D = 121
Since:
D > 0
the equation has two distinct real roots.
The garden dimensions are therefore possible.
Now solve:
x = (−5 ± √121) ÷ 2
x = (−5 ± 11) ÷ 2
x₁ = 3
x₂ = −8
Width cannot be negative.
Therefore:
- Width = 3 m
- Length = 8 m
Area = 24 m²
Understanding the Quadratic Formula
Quadratic formula:
x = (−b ± √D) ÷ 2a
This produces two roots:
x₁ = (−b + √D) ÷ 2a
x₂ = (−b − √D) ÷ 2a
Observe what happens for different values of D:
- D > 0 → square root exists → two different roots
- D = 0 → square root becomes zero → equal roots
- D < 0 → square root of a negative number → no real roots
Worked Examples
Example 1: Two Distinct Roots
Equation:
2x² − 7x + 3 = 0
a = 2
b = −7
c = 3
D = (−7)² − 4 × 2 × 3
D = 49 − 24
D = 25
Since D > 0, two distinct real roots exist.
Example 2: Equal Roots
Equation:
x² − 6x + 9 = 0
D = (−6)² − 4 × 1 × 9
D = 36 − 36
D = 0
Therefore, equal roots exist.
Root:
x = −b ÷ 2a
x = 6 ÷ 2
x = 3
Example 3: No Real Roots
Equation:
x² + x + 1 = 0
D = 1² − 4 × 1 × 1
D = −3
Since D < 0, no real roots exist.
Visual Meaning of the Discriminant
- D > 0 → parabola intersects the x-axis twice
- D = 0 → parabola touches the x-axis once
- D < 0 → parabola does not intersect the x-axis
Common Mistakes Students Make
Mistake 1: Miscalculating 4ac
Students sometimes calculate 4ac incorrectly.
Remember:
4ac = 4 × a × c
Calculate the complete product carefully.
Mistake 2: Confusing D = 0 with No Roots
Correct rules:
- D = 0 → equal real roots
- D < 0 → no real roots
Mistake 3: Forgetting Standard Form
Always convert equations into:
ax² + bx + c = 0
before identifying coefficients.
Important Rules for CBSE Exams
| Value of D | Nature of Roots |
|---|---|
| D > 0 (perfect square) | Two distinct rational roots |
| D > 0 (not a perfect square) | Two distinct irrational roots |
| D = 0 | Two equal real roots |
| D < 0 | No real roots |
Types of Questions CBSE Asks
- Find the discriminant and state the nature of roots
- Find the value of k for equal roots
- Find the value of k for real roots
- Find the value of k for distinct roots
Real-World Applications
Projectile Motion
Height equations in physics often form quadratic equations.
The discriminant helps determine whether an object reaches a certain height.
Geometry Problems
Rectangles, areas, and dimensions frequently create quadratic equations.
The discriminant checks whether dimensions are geometrically possible.
Engineering and Physics
Quadratic equations appear in:
- Motion equations
- Structural calculations
- Optimisation problems
The discriminant predicts whether solutions exist.
Key Idea in Simple Words
A quadratic equation can have:
- Two real roots
- One repeated root
- No real roots
The value of D = b² − 4ac reveals which type of roots the equation contains.
It allows mathematicians to predict the nature of roots before solving the equation.
Practice Questions
- Find the discriminant of 3x² − 5x + 2 = 0 and state the nature of its roots.
- For what value of k does kx² + 6x + 1 = 0 have equal roots?
- Show that x² + 5x + 7 = 0 has no real roots.
- A rectangular park has perimeter 46 m and area 120 m². Form a quadratic equation and test whether the park dimensions are possible.
- If α and β are roots of 2x² − 5x + 3 = 0, find α + β and αβ.
Frequently Asked Questions
Why is it called the discriminant?
Because it distinguishes between different types of roots.
Does the discriminant solve the equation?
No.
It only predicts the nature of roots.
Why can a quadratic have only two roots?
A polynomial of degree 2 always has two roots in the complex number system.
What does D = 0 mean geometrically?
The parabola touches the x-axis exactly once.
Are equations with D < 0 unsolvable?
They have no real roots, but they do have complex roots studied in higher classes.
Related Topics
- Factorisation of Quadratic Equations
- Completing the Square Method
- Relationship Between Zeros and Coefficients
This article follows the CBSE Class 10 Mathematics syllabus (Chapter 4: Quadratic Equations) and aligns with NCERT Class 10 Mathematics concepts. All worked examples are mathematically verifiable and designed to reflect common CBSE examination patterns.
About Author Dhruv Chandravanshi
Dhruv Chandravanshi writes about physical and logical systems, from motion to matrices, uncovering the structures that govern how matter and numbers behave.
