📘 Quadratic Equations Class 10 Notes (Chapter 4)

🔷 Introduction

Quadratic equations are an important topic in algebra. In simple words, a quadratic equation is an equation in which the highest power of the variable is 2.

For example:$$x^2 + 5x + 6 = 0$$x2+5x+6=0

Therefore, every equation with degree 2 is called a quadratic equation.

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🔹 Standard Form of Quadratic Equation

The standard form of a quadratic equation is:$$ax^2 + bx + c = 0$$ax2+bx+c=0

Here:

• a, b, c are real numbers
• a ≠ 0

Moreover, if a becomes zero, the equation will no longer be quadratic.

🔹 Examples of Quadratic Equations

Some common examples are:

• $x^2 + 3x + 2 = 0$x2+3x+2=0
• $2x^2 + 7x + 5 = 0$2×2+7x+5=0
• $5x^2 – 4x + 1 = 0$5×2−4x+1=0

🔷 Methods to Solve Quadratic Equations

There are three main methods.

🔹 1. Factorization Method

In this method, we split the middle term and find factors.

👉 Example:$$x^2+5x+6=0$$x2+5x+6=0

Factorizing:$$x^2+2x+3x+6=0$$x2+2x+3x+6=0 $$x(x+2)+3(x+2)=0$$x(x+2)+3(x+2)=0 $$(x+2)(x+3)=0$$(x+2)(x+3)=0

So,$$x=-2,\quad x=-3$$x=−2,x=−3

Hence, these are the roots.

🔹 2. Completing the Square Method

In this method, we make a perfect square.

👉 Example:$$x^2+6x+5=0$$x2+6x+5=0 $$x^2+6x=-5$$x2+6x=−5

Add 9 on both sides:$$x^2+6x+9=4$$x2+6x+9=4 $$(x+3)^2=4$$(x+3)2=4 $$x+3=\pm2$$x+3=±2

So:$$x=-1,\quad x=-5$$x=−1,x=−5

🔹 3. Quadratic Formula Method

The formula is:$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$x=2a−b±b2−4ac​​

Therefore, this method works for every quadratic equation.

🔷 Discriminant

The value inside the square root is called the discriminant.$$D=b^2-4ac$$D=b2−4ac

The discriminant tells us about the nature of roots.

🔹 Nature of Roots

✔ If D > 0

Two distinct real roots.

✔ If D = 0

Two equal real roots.

✔ If D < 0

No real roots.

Thus, the discriminant helps us understand the solution quickly.

🔷 Roots of Quadratic Equation

If α and β are roots of:$$ax^2+bx+c=0$$ax2+bx+c=0

Then:

Sum of roots:

$$\alpha+\beta=-\frac{b}{a}$$α+β=−ab​

Product of roots:

$$\alpha\beta=\frac{c}{a}$$αβ=ac​