Mastering Quadratic Equations 📘

📚 1. Introduction and Core Concepts

A quadratic equation in one variable is an equation that can be written in the standard form:

where:

• $a, b,$ and $c$ are real numbers
• $a \neq 0$
• $x$ is the variable

If $a = 0$, the equation becomes linear, not quadratic.

Degree of the equation

The highest power of the variable is 2, so it is called a quadratic (degree 2) equation.

Examples of quadratic equations

• $3x^2 - 5x + 2 = 0$
• $x^2 - 9 = 0$
• $2x^2 + 7x = 0$

All these can be written in the form $ax^2 + bx + c = 0$ with $a \neq 0$.

Not quadratic

• $5x + 3 = 0$ (degree 1: linear)
• $x^3 - 4x + 1 = 0$ (degree 3: cubic)

Roots / Solutions of a Quadratic Equation

The roots or solutions of a quadratic equation $ax^2 + bx + c = 0$ are the values of $x$ which satisfy the equation (make the LHS zero).

If $\alpha$ and $\beta$ are the roots of the equation, then:

Number of roots (solutions) of a quadratic equation: at most 2.

🔍 2. Detailed Breakdown & Classifications

Below is a quick view of some important terms related to quadratic equations:

2.1 Different Methods of Solving Quadratic Equations

At Class 10 level, you need to master three main methods:

1. Factorisation method
2. Completing the square method (conceptual, used less in Board questions)
3. Quadratic formula

(A) Solving by Factorisation

General idea: Convert the quadratic equation to a product of two binomials and use the zero product rule.

Example:

Step 1: Split the middle term (−5x) into two terms whose sum is −5 and product is $1 \times 6 = 6$.

Numbers are −2 and −3.

Group and factor:

Take common factor:

So, either $x - 3 = 0$ or $x - 2 = 0$

Thus, $x = 3$ or $x = 2$.

These are the roots.

Factorisation works best when the quadratic is easily factorable.

(B) Solving by Completing the Square (Concept)

This method makes a perfect square on one side.

Example:

Move constant term:

Add $\left(\frac{4}{2}\right)^2 = 4$ to both sides:

LHS is a perfect square:

Taking square root on both sides:

So:

This method is the base for deriving the quadratic formula.

(C) Quadratic Formula

For the quadratic equation:

The roots are given by:

This is the most powerful method and works for any quadratic equation (even when it is not factorable).

2.2 Discriminant and Nature of Roots

The expression:

is called the discriminant.

1. If $D > 0$:

   • Two real and distinct roots

2. If $D = 0$:

   • Two real and equal roots (repeated root)

3. If $D < 0$:

   • No real roots (complex roots – not studied in detail in Class 10)

Example

Consider:

Here, $a = 1, b = -4, c = 3$

So, roots are real and distinct.

2.3 Sum and Product of Roots

If $\alpha$ and $\beta$ are roots of $ax^2 + bx + c = 0$, then:

These relations are very useful in MCQs and in questions where you need to form a quadratic equation from given roots.

Example

For the equation:

$a = 2, b = -5, c = 3$

Sum of roots:

Product of roots:

⚙️ 3. Essential Rules, Formulas, or Mechanisms

Here are the key formulas and results you must remember for CBSE Class 10 exams.

(1) Standard form of a quadratic equation

(2) Quadratic formula

(3) Discriminant

• $D > 0$ → real and distinct roots
• $D = 0$ → real and equal roots
• $D < 0$ → no real roots

(4) Sum and Product of roots

For roots $\alpha$ and $\beta$ of $ax^2 + bx + c = 0$:

(5) Form of quadratic equation when roots are known

If roots are $\alpha$ and $\beta$, then the quadratic equation (with leading coefficient 1) is:

More generally (with coefficient $k$):

Real-Life Application Example (Very Important for Word Problems)

Problem idea: The product of two consecutive natural numbers is 132. Find the numbers.

Let the first number be $n$. Then the next consecutive number is $n + 1$.

Product condition:

This is a quadratic equation in $n$.

Now solve using factorisation or quadratic formula.

Factorising:

We need two numbers whose product is $-132$ and sum is $1$.

Numbers are $12$ and $-11$.

So, $n = 11$ or $n = -12$

Since natural numbers are positive, $n = 11$

Numbers are 11 and 12.

📝 4. Summary & Conclusion

• A quadratic equation is any equation that can be written in the form $ax^2 + bx + c = 0$ with $a \neq 0$.
• It has at most two roots (solutions).
• You can solve quadratic equations by:
  • Factorisation
  • Completing the square (mainly for understanding)
  • Quadratic formula

The quadratic formula:

is the most general and powerful method.

• The discriminant $D = b^2 - 4ac$ tells you the nature of roots:

  • $D > 0$: two real and distinct roots
  • $D = 0$: real and equal roots
  • $D < 0$: no real roots

• Sum and product of roots of $ax^2 + bx + c = 0$:

  • Sum = $-\dfrac{b}{a}$
  • Product = $\dfrac{c}{a}$

These are very useful for:

• Forming a quadratic equation when roots are given
• Quick checking and MCQ-type questions

From the exam point of view, focus on:

• Converting word problems into correct quadratic equations
• Choosing a suitable solving method
• Carefully handling signs and arithmetic in the discriminant and quadratic formula

If you practise a variety of problems — especially from NCERT textbook and exemplar — quadratic equations can easily become a high-scoring chapter for your Class 10 Maths Board exam.