📘 CLASS 10 – QUADRATIC EQUATIONS

(DETAILED NOTES – NCERT BASED)

---

1️⃣ What is a Quadratic Equation?

An equation of the form ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 where

• a,b,ca, b, ca,b,c are real numbers
• a≠0a \neq 0a=0

is called a quadratic equation in one variable.

Why “Quadratic”?

Because the highest power of the variable is 2.

---

2️⃣ Examples of Quadratic Equations

✔ x2+7x+10=0x^2 + 7x + 10 = 0x2+7x+10=0
✔ 2x2−3x+5=02x^2 – 3x + 5 = 02×2−3x+5=0
✔ −4x2+9=0-4x^2 + 9 = 0−4×2+9=0

❌ Not quadratic:

• x3+2x+1=0x^3 + 2x + 1 = 0x3+2x+1=0 (degree 3)
• 5x−2=05x – 2 = 05x−2=0 (linear)

---

3️⃣ Standard Form of Quadratic Equation

ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0

Where:

• aaa → coefficient of x2x^2x2
• bbb → coefficient of xxx
• ccc → constant term

⚠️ If a=0a = 0a=0, the equation is not quadratic.

---

4️⃣ Roots (Solutions) of a Quadratic Equation

The values of x that satisfy the equation are called its roots.

A quadratic equation always has two roots
(these roots may be real or imaginary).

---

5️⃣ Methods of Solving Quadratic Equations

There are three methods prescribed in Class 10:

---

🔹 Method 1: Factorisation Method

When to use?

✔ When the equation can be factorised easily.

Steps:

1. Write the equation in standard form.
2. Split the middle term.
3. Factorise.
4. Set each factor equal to zero.
5. Solve.

Example:

Solve: x2+5x+6=0x^2 + 5x + 6 = 0x2+5x+6=0x2+3x+2x+6=0x^2 + 3x + 2x + 6 = 0x2+3x+2x+6=0 (x+3)(x+2)=0(x + 3)(x + 2) = 0(x+3)(x+2)=0 x=−3, −2x = -3,\ -2x=−3, −2

---

🔹 Method 2: Completing the Square Method

Important Concept:

Any expression of the form
(x+a)2=x2+2ax+a2(x + a)^2 = x^2 + 2ax + a^2(x+a)2=x2+2ax+a2

Steps:

1. Make coefficient of x2=1x^2 = 1x2=1
2. Move constant term to RHS
3. Add square of half of coefficient of x
4. Write LHS as a perfect square
5. Solve

Example:

Solve: x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0x2+6x=−5x^2 + 6x = -5x2+6x=−5

Add 999 to both sides:(x+3)2=4(x + 3)^2 = 4(x+3)2=4 x+3=±2x + 3 = \pm 2x+3=±2 x=−1, −5x = -1,\ -5x=−1, −5

---

🔹 Method 3: Quadratic Formula (MOST IMPORTANT)

Formula:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}x=2a−b±b2−4ac​​

This method always works, even when factorisation fails.

---

Example:

Solve: 3x2−5x−2=03x^2 – 5x – 2 = 03×2−5x−2=0

Here:

• a=3a = 3a=3
• b=−5b = -5b=−5
• c=−2c = -2c=−2

x=5±25+246x = \frac{5 \pm \sqrt{25 + 24}}{6}x=65±25+24​​ x=5±76x = \frac{5 \pm 7}{6}x=65±7​ x=2, −13x = 2,\ -\frac{1}{3}x=2, −31​

---

6️⃣ Discriminant

The expressionD=b2−4acD = b^2 – 4acD=b2−4ac

is called the discriminant.

Importance:

The discriminant tells us the nature of roots without solving.

---

Nature of Roots Table:

Value of D	Nature of Roots
D>0D > 0D>0	Two real & distinct roots
D=0D = 0D=0	Two real & equal roots
D<0D < 0D<0	No real roots

---

Example:

For x2−4x+4=0x^2 – 4x + 4 = 0x2−4x+4=0D=(−4)2−4(1)(4)=0D = (-4)^2 – 4(1)(4) = 0D=(−4)2−4(1)(4)=0

Roots are real and equal.

---

7️⃣ Relationship Between Roots and Coefficients

Let roots be α\alphaα and β\betaβ

For ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0:α+β=−ba\alpha + \beta = \frac{-b}{a}α+β=a−b​ αβ=ca\alpha \beta = \frac{c}{a}αβ=ac​

---

8️⃣ Forming a Quadratic Equation

If sum and product of roots are given:x2−(α+β)x+αβ=0x^2 – (\alpha + \beta)x + \alpha\beta = 0x2−(α+β)x+αβ=0

Example:

Roots: 3 and 5x2−8x+15=0x^2 – 8x + 15 = 0x2−8x+15=0

---

9️⃣ Applications of Quadratic Equations (Word Problems)

Common topics:
✔ Consecutive integers
✔ Area & dimensions
✔ Speed, distance, time
✔ Age problems

---

Example:

The product of two consecutive positive integers is 156.

Let numbers be xxx and x+1x+1x+1x(x+1)=156x(x+1) = 156x(x+1)=156 x2+x−156=0x^2 + x – 156 = 0x2+x−156=0

Solve using quadratic formula.

---

🔟 Important Exam Points ⭐

✔ Always write standard form
✔ Mention values of a, b, c
✔ Show discriminant clearly
✔ Use quadratic formula if factorisation is difficult
✔ Do not forget ± sign

---

📌 Summary

• Quadratic equation → degree 2
• Standard form → ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0
• Three solving methods
• Discriminant decides nature of roots
• Very important chapter for board exams