Quadratic equation

Quadratic Equation – Standard Form and Explanation

An equation of the form ax² + bx + c = 0

is called a quadratic equation, where:

- a ≠ 0

- a, b, and c are real numbers

- The degree of the equation is 2 (because the highest power of x is 2)

Why should 'a' not be zero?

Let’s understand why the condition a ≠ 0 is important.

If we take a = 0, then the equation becomes:

    0x² + bx + c = 0

    ⇒ bx + c = 0

Now, the equation has become a linear equation, because the highest degree of x is now 1.

Hence, it is not a quadratic equation anymore.

Can b or c be zero?

Yes, b and/or c can be zero in a quadratic equation.

Here are a few examples:

- If b = 0: The equation becomes ax² + c = 0, Degree of the equation is 2

- If c = 0: The equation becomes ax² + bx = 0 ,Degree of the equation is 2

- If both b = 0 and c = 0: The equation becomes ax² = 0, Degree of the equation is 2

In all these cases, the equation still remains a quadratic equation as long as a ≠ 0.

Conclusion

To be a quadratic equation, the coefficient of x² (i.e., 'a') must not be zero (a ≠ 0).

However, b and c can be zero, and the equation will still be quadratic as long as a ≠ 0.

                                           HINGLISH EXPLANATION

Quadratic Equation – Standard Form aur Explanation (Hinglish)

Ek equation jo is form mein hoti hai ax² + bx + c = 0

use Quadratic Equation kehte hain, jahan:

- a ≠ 0 hona chahiye

- a, b, aur c real numbers hote hain

- Equation ki degree 2 hoti hai kyunki highest power of x = 2 hai

a ≠ 0 kyun hona chahiye?

Chaliye samajhte hain ki a ≠ 0 hona kyon zaroori hai.

Agar hum a = 0 le lein, toh equation ban jaati hai:

    0x² + bx + c = 0

    ⇒ bx + c = 0

Ab equation ka degree 1 ho gaya hai, kyunki x² wala term gayab ho gaya.

Isliye yeh equation ab linear equation ban gayi hai, quadratic nahi.

Kya b ya c zero ho sakte hain?

Haan, b ya c zero ho sakte hain aur equation fir bhi quadratic hi rahegi.

Kuch examples dekhte hain:

- Agar b = 0 ho: Equation ban jaati hai: ax² + c = 0. Degree ab bhi 2 hai.

- Agar c = 0 ho: Equation ban jaati hai: ax² + bx = 0. Degree ab bhi 2 hai.

- Agar b = 0 aur c = 0 ho: Equation ban jaati hai: ax² = 0. Degree ab bhi 2 hai.

In sabhi cases mein, jab tak a ≠ 0 hai, equation quadratic hi rahegi.

Conclusion (Nishkarsh)

Ek quadratic equation ke liye x² ka coefficient 'a' zero nahi hona chahiye (a ≠ 0).

Lekin b aur c zero ho sakte hain, fir bhi equation quadratic hi rahegi jab tak a ≠ 0.