Class 10 Mathematics – Quadratic Equations | Interactive Visual Revision Notes

📝 Section 1 — Classroom Notes

🔢 Topic 1: What Is a Quadratic Equation?

🌎 Real World Connection: A ball thrown upward follows the path h = −5t² + 20t. To find when it lands (h = 0), we solve a quadratic equation. Architects use quadratics to design arches and domes. Car engineers calculate braking distances using quadratic equations. Even the shape of a satellite dish is a parabola — the curve described by a quadratic.

It is an equation with x² as the highest power.
The word quadratic comes from "quad" = square.
Example: x² + 3x + 2 = 0
It always has degree 2.
It can have at most 2 solutions.

🔢 Topic 2: Standard Form

Standard form: ax² + bx + c = 0
Here a, b, c are real numbers.
The value of a cannot be 0.
If a = 0, it becomes a linear equation.
a = coefficient of x², b = coefficient of x, c = constant.

🔢 Topic 3: Roots of a Quadratic Equation

A root is a value of x that satisfies the equation.
If we put the root in place of x, we get 0.
Roots are also called zeroes or solutions.
A quadratic can have 2, 1, or 0 real roots.
Example: x = 2 is a root of x² − 5x + 6 = 0.

🔢 Topic 4: Solving by Factorisation

Split the equation into two brackets.
Find two numbers that multiply to give ac.
Those same two numbers must add to give b.
Set each bracket equal to zero.
Solve each bracket to find the two roots.

🔢 Topic 5: Completing the Square

Move the constant c to the right side.
Divide everything by a if a ≠ 1.
Add (b/2a)² to both sides.
The left side becomes a perfect square.
Take square root of both sides and solve.

🔢 Topic 6: The Quadratic Formula

The formula gives roots directly.
x = (−b ± √(b² − 4ac)) ÷ 2a
It works for every quadratic equation.
Just put values of a, b, c and calculate.
The ± sign gives us two answers.

🔢 Topic 7: The Discriminant

The discriminant is D = b² − 4ac.
It is the part under the square root.
D tells us the nature of roots.
D is also written as Δ (delta).
It decides how many roots are real.

🔢 Topic 8: Nature of Roots

If D > 0 → Two different real roots.
If D = 0 → Two equal real roots.
If D < 0 → No real roots.
D > 0 and a perfect square → rational roots.
D > 0 but not a perfect square → irrational roots.

🔢 Topic 9: Sum and Product of Roots

If α and β are the two roots:
Sum of roots = α + β = −b/a
Product of roots = α × β = c/a
These formulas save time in many problems.
No need to find individual roots every time.

🔢 Topic 10: Forming Equations from Roots

If roots are α and β, the equation is:
x² − (α+β)x + αβ = 0
Find sum and product of the given roots.
Put them into the formula above.
Example: roots 3 and −5 → x² + 2x − 15 = 0

🎨 Section 2 — Easy Illustrations

Picture 1 — Breaking down ax² + bx + c = 0

Picture 2 — Four steps to solve by factorisation

Picture 3 — The parabola crosses the x-axis at the roots

Picture 4 — D decides if the curve crosses, touches, or misses the x-axis

Picture 5 — Every part of the quadratic formula explained

✏️ Section 3 — Worked Examples

✏️ Example 1: Solve by Factorisation

Solve: 3x² − 11x + 6 = 0

1 Here a = 3, b = −11, c = 6

2 Find a × c = 3 × 6 = 18

3 Find two numbers: product = 18, sum = −11

4 The numbers are −9 and −2

5 Split: 3x² − 9x − 2x + 6 = 0

6 Group: 3x(x − 3) − 2(x − 3) = 0

7 Factor: (3x − 2)(x − 3) = 0

8 3x − 2 = 0 → x = 2/3 OR x − 3 = 0 → x = 3

🎯 x = 2/3 or x = 3

✏️ Example 2: Solve Using the Quadratic Formula

Solve: 2x² + 5x − 3 = 0

1 Here a = 2, b = 5, c = −3

2 D = b² − 4ac = 25 − 4(2)(−3) = 25 + 24 = 49

3 √D = √49 = 7

4 x = (−b ± √D) ÷ 2a = (−5 ± 7) ÷ 4

5 x = (−5 + 7) ÷ 4 = 2 ÷ 4 = 1/2

6 x = (−5 − 7) ÷ 4 = −12 ÷ 4 = −3

🎯 x = 1/2 or x = −3

✏️ Example 3: Find the Nature of Roots

Find the nature of roots: 4x² − 12x + 9 = 0

1 Here a = 4, b = −12, c = 9

2 D = b² − 4ac = (−12)² − 4(4)(9)

3 D = 144 − 144 = 0

4 D = 0 means two equal real roots.

5 x = −b ÷ 2a = 12 ÷ 8 = 3/2

🎯 Two equal roots: x = 3/2 and x = 3/2

📊 Section 4 — Cheat Sheet

Concept	Formula / Rule	Example
Standard Form	ax² + bx + c = 0, a ≠ 0	5x² + 3x − 7 = 0
Quadratic Formula	x = (−b ± √D) ÷ 2a	Always works!
Discriminant	D = b² − 4ac	Tells nature of roots
D > 0	Two different real roots	Parabola cuts x-axis twice
D = 0	Two equal real roots	Parabola touches x-axis once
D < 0	No real roots	Parabola misses x-axis
Sum of roots	α + β = −b/a	Quick shortcut
Product of roots	α × β = c/a	Quick shortcut
Form equation	x² − (sum)x + (product) = 0	From given roots
Factorisation	Split middle term, group, solve	Fastest when it works

📄 Section 5 — Take-Home Handout

✍️ A. True or False

1. Every quadratic has exactly 2 real roots. ___
2. If D = 0, both roots are equal. ___
3. The quadratic formula works only when D > 0. ___
4. Sum of roots = −b/a. ___
5. If a = 0, the equation is still quadratic. ___

✍️ B. Fill in the Blanks

1. The discriminant D = ________.
2. If D < 0, there are ________ real roots.
3. Product of roots = ________.
4. The highest power in a quadratic is ________.
5. α + β = ________ (in terms of a, b).

✍️ C. Solve These

1. Solve by factorisation: x² − 7x + 10 = 0

Show your work here

2. Find D and nature of roots: 5x² − 4x + 1 = 0

Show your work here

🔗 D. Match the Columns

Column A	Column B
1. D > 0	a. No real roots
2. D = 0	b. Two different roots
3. D < 0	c. −b/a
4. Sum of roots	d. Two equal roots

🎨 E. Form the Equation

Form a quadratic equation whose roots are 4 and −7.

Show your work here

✅ Detailed Solutions

A. True or False — Answers

1 FALSE — If D < 0, there are no real roots. A quadratic need not have 2 real roots.

2 TRUE — If D = 0 both roots are equal (repeated root).

3 FALSE — The quadratic formula x = (−b ± √D)/2a works for all values of D.

4 TRUE — Sum of roots = −b/a is correct by Vieta’s formulas.

5 FALSE — If a = 0 the x² term vanishes; it becomes linear, not quadratic.

B. Fill in the Blanks — Answers

1 The discriminant D = b² − 4ac.

2 If D < 0, there are no real roots.

3 Product of roots = c/a.

4 The highest power in a quadratic is 2.

5 α + β = −b/a.

C1. Factorise x² − 7x + 10 = 0

1 Find two numbers: product = 10, sum = −7 → −5 and −2

2 x² − 7x + 10 = (x − 5)(x − 2) = 0

3 Either x − 5 = 0 → x = 5, or x − 2 = 0 → x = 2

x = 2 or x = 5

C2. Nature of roots: 5x² − 4x + 1 = 0

1 a = 5, b = −4, c = 1

2 D = b² − 4ac = (−4)² − 4(5)(1) = 16 − 20 = −4

3 D < 0 → No real roots

D = −4 < 0 : No real roots exist

D. Match the Columns — Answers

1 D > 0 → b (Two different real roots)

2 D = 0 → d (Two equal real roots)

3 D < 0 → a (No real roots)

4 Sum of roots → c (= −b/a)

E. Equation with roots 4 and −7

1 Sum of roots: 4 + (−7) = −3

2 Product of roots: 4 × (−7) = −28

3 Equation: x² − (sum)x + (product) = 0

4 x² − (−3)x + (−28) = 0 → x² + 3x − 28 = 0

x² + 3x − 28 = 0

🃏 Section 6 — Revision Flashcards

👉

🔢

What is the standard form of a quadratic equation?

ax² + bx + c = 0

where a ≠ 0

🔎

What is the quadratic formula?

x = (−b ± √(b² − 4ac)) ÷ 2a

📈

What is the discriminant?

D = b² − 4ac

It tells us the nature of roots.

❓

What happens when D > 0?

The equation has two different real roots.

The parabola crosses the x-axis twice.

💡

What happens when D = 0?

The equation has two equal roots.

The parabola just touches the x-axis.

🔧

What is the sum of roots formula?

α + β = −b/a

🎯

What is the product of roots formula?

α × β = c/a

🔗

How to form an equation from roots?

x² − (sum)x + (product) = 0

Put sum and product of roots.

Class 10 Mathematics — Quadratic Equations

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🔢 Class 10 Mathematics — Quadratic Equations Interactive Visual Revision Notes Aligned with the Class 10 Mathematics syllabus. Independently created study material. Not affiliated with or endorsed by NCERT.

📋 What is Inside?

📝 Section 1 — Classroom Notes

🔢 Topic 1: What Is a Quadratic Equation?

🔢 Topic 2: Standard Form

🔢 Topic 3: Roots of a Quadratic Equation

🔢 Topic 4: Solving by Factorisation

🔢 Topic 5: Completing the Square

🔢 Topic 6: The Quadratic Formula

🔢 Topic 7: The Discriminant

🔢 Topic 8: Nature of Roots

🔢 Topic 9: Sum and Product of Roots

🔢 Topic 10: Forming Equations from Roots

🎨 Section 2 — Easy Illustrations

✏️ Section 3 — Worked Examples

✏️ Example 1: Solve by Factorisation

✏️ Example 2: Solve Using the Quadratic Formula

✏️ Example 3: Find the Nature of Roots

📊 Section 4 — Cheat Sheet

📄 Section 5 — Take-Home Handout

✍️ A. True or False

✍️ B. Fill in the Blanks

✍️ C. Solve These

🔗 D. Match the Columns

🎨 E. Form the Equation

✅ Detailed Solutions

A. True or False — Answers

B. Fill in the Blanks — Answers

C1. Factorise x² − 7x + 10 = 0

C2. Nature of roots: 5x² − 4x + 1 = 0

D. Match the Columns — Answers

E. Equation with roots 4 and −7

🃏 Section 6 — Revision Flashcards

🧠 Section 7 — Mind Map