2nd Year math ch # 1 – Ex: 1.1 Notes – New Book 2026-27

Bunyadi Concept (Method of Factorization)

Jab hum kisi quadratic function (parabola) ka graph banana chahte hain, to sabse asaan tareeqa ye hai:

1. Roots (x-intercepts) nikalo — yani wo points jahan graph x-axis ko cut karta hai (y = 0 rakh kar).
2. Vertex nikalo — dono roots ka midpoint lo, wahi vertex ka x-coordinate hota hai. Phir us x ko function mein daal kar y-coordinate nikal lo.
3. Direction dekho — agar leading coefficient (a) positive hai to parabola upar ki taraf khulta hai (upward), agar negative hai to neeche ki taraf (downward).

Question 1 — Har Part Ki Explanation

(i) y = 3(x+1)(x−1)

• Ye already factorized hai, isliye roots seedhe nikal aaye: x = −1 aur x = 1.
• Leading coefficient a = 3, jo positive hai → parabola upward khulega.
• Vertex ka x = roots ka average = (−1+1)/2 = 0. Phir x=0 rakh kar y = 3(1)(−1) = −3 mila.
• Vertex = (0, −3), jo yahi y-intercept bhi hai.

(ii) f(x) = −2(x−1)(x−2)

• Roots: x = 1, x = 2.
• a = −2 (negative) → downward khulega.
• Vertex x = (1+2)/2 = 1.5. Ye value function mein daal kar y = 0.5 mila.
• y-intercept ke liye x=0 rakha: f(0) = −4.

(iii) y = (2x−1)(2x+1)

• Roots nikalne ke liye har bracket ko zero karo: 2x−1=0 → x=1/2, aur 2x+1=0 → x=−1/2.
• Expand karne par y = 4x² − 1 banta hai, isliye a = 4 (positive) → upward.
• Vertex x=0 (kyunke roots symmetric hain −1/2 aur 1/2 ke), y = −1.

(iv) y = 2x² + x − 3

• Ye already factorized form mein nahi diya, isliye pehle factorize karna parta hai.
• Trick: a×c = 2×(−3) = −6 nikalo, aur wo do numbers dhoondo jo multiply karke −6 den aur add karke b=1 den → wo hain 3 aur −2.
• Phir middle term ko split karke group banate hain: 2x²+3x−2x−3 = (2x+3)(x−1).
• Roots: x = −3/2, x = 1.
• a=2 (positive) → upward. Vertex x = (−3/2+1)/2 = −1/4, jahan y = −25/8 nikla (calculation thora lamba hai lekin same tareeqa).

(v) f(x) = x² + 2x + 1

• Ye ek perfect square hai: (x+1)².
• Jab bracket square hoti hai to root repeated hoti hai (yani x=−1 do dafa aata hai).
• Iska matlab graph x-axis ko cross nahi karta, balke sirf touch karta hai us point par aur wapas mud jata hai.
• a=1 (positive) → upward.

(vi) y = 3(x−1)²

• Same jaisa (v) — perfect square, repeated root x=1.
• Graph x-axis ko x=1 par touch karega, cross nahi karega.
• a=3 → upward.

Important Concept: Agar function perfect square [(x−k)² jaisi form] mein ho, to uska sirf ek hi root hota hai (repeated), aur graph us point par x-axis ko sirf chhoo ke wapas chala jata hai — cross nahi karta.

Question 2 — Graph Dekh Kar Equation Banana (Ulta Kaam)

Yahan Question 1 ka ulta karna hai — humein graph diya gaya hai, aur us se equation nikalni hai.

(i) Graph x-axis ko x=−2 aur x=1 par cut karta hai (do alag roots), isliye form banayi:

$$y=a(x+2)(x-1)$$

Ab humein a ki value nahi pata, isliye graph se ek aur point liya — (3/2, −7/4) — aur wo equation mein substitute karke a nikala: a = −1.
Final: y = −(x+2)(x−1)

(ii) Graph x=2 par sirf touch karta hai (cross nahi karta) → matlab ye repeated root hai:

y=a(x−2)2y = a(x-2)^2y=a(x−2)2

Phir point (1,1) substitute karke a=1 nikala.
Final: y = (x−2)²

Concept ye hai: Agar graph x-axis ko cross karta hai to form (x−r₁)(x−r₂) use hoti hai (do alag roots). Agar sirf touch karta hai to form (x−r)² use hoti hai (ek repeated root).

Question 3, 4, 5 — Diye Gaye Conditions Se Equation Banana

Ye teeno questions same logic follow karte hain:

Q3: x-intercepts −1 aur −2 diye, aur ek point (−3,2) diya.
→ Form: y = a(x+1)(x+2), phir point substitute karke a=1 mila.

Q4: x=−1 par touch karta hai (matlab repeated root), aur y-intercept −2 diya.
→ Form: y = a(x+1)², phir x=0,y=−2 substitute karke a=−2 mila.

Q5: x-intercepts −1 aur 1, vertex (0,−2) diya.
→ Form: y = a(x+1)(x−1), vertex ka x=0 already roots ke midpoint se match ho raha tha, isliye seedha x=0,y=−2 substitute karke a=2 mila.

Sabse Zaroori Nuqta (Summary for Students)

Ye tareeqa har question mein repeat ho raha hai — bas farq itna hai ke kabhi hum root se equation bana rahe hain (Q1), aur kabhi diye gaye info se equation reverse-engineer kar rahe hain (Q2–Q5).