Quadratic Equation (দ্বিঘাত সমীকৰণ): Different Types

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1. Standard Form Quadratic Equation (দ্বিঘাত সমীকৰণ)

Question / প্ৰশ্ন: Solve: 2x² + 7x + 3 = 0

Solution / সমাধান:

1st: Product = 2 × 3 = 6, Sum = 7 → 6 and 1

2nd: Rewrite: 2x² + 6x + x + 3 = 0

3rd: Group terms: Group terms: 2x (x+3) + 1(x+3) = 0 

4th: Factorize: (x+3)(2x+1) = 0

5th: x + 3 = 0 ⇒ x = −3 
2x + 1 = 0 ⇒ x = −1/2​

Ans / উত্তৰ: x = −3,  −1/2 

2. Pure Quadratic Equation (বিশুদ্ধ দ্বিঘাত সমীকৰণ)

Question / প্ৰশ্ন: Solve: 5x² − 80 = 0

Solution / সমাধান:

1st: 5x² = 80

2nd: Divide by 5 → x² = 16

3rd: Take square root → x = ±√16​

4th: x = ±4

Ans / উত্তৰ: x = ±4

3. Incomplete Quadratic Equation (অসম্পূৰ্ণ দ্বিঘাত সমীকৰণ)

Question / প্ৰশ্ন: Solve: x² − 9x = 0

Solution / সমাধান:

1st: Factor out common term → x (x − 9) = 0

2nd: Apply zero-product rule: x = 0 or x = 9

Ans / উত্তৰ: x = 0, 9

4. Factorable Quadratic Equation (গুণনীয়ক পদ্ধতিৰে দ্বিঘাত সমীকৰণ)

Question / প্ৰশ্ন: Solve: x² − 11x + 24 = 0

Solution / সমাধান:

1st: Product = 24, Sum = –11 → –3 and –8

2nd: Factorize: (x−3)(x−8) = 0

3rd: x − 3 = 0 ⇒ x = 3
x − 8 = 0 ⇒ x = 8

Ans / উত্তৰ: x = 3,  8

9. Quadratic Equation with Repeated Roots (পুনৰাবৃত্ত মূলযুক্ত দ্বিঘাত সমীকৰণ)

Question: Solve: x² − 14x + 49 = 0

Solution:

1st: x² - 14x + 49 = (x-7)²

2nd: (x-7)² = 0

3rd: x - 7 = 0

4th: x = 7

Ans: x = 7