Quadratic equations — factoring, completing the square

2.1.16 · Maths › Algebra — Introduction & Intermediate

Core insight yeh hai: Hum $a x^{2} + b x + c = 0$ ko ek product $(x - p) (x - q) = 0$ ya ek perfect square $(x - h)^{2} = k$ mein transform kar rahe hain, kyunki agar koi product zero ke barabar hai, toh kam se kam ek factor zero hona chahiye. Yeh ek polynomial problem ko simple linear equations mein badal deta hai.

Method 1: Factoring

Factor Kaise Karein (Step-by-Step Derivation)

Case 1: Monic quadratics ( $a = 1$ , toh hamare paas $x^{2} + b x + c = 0$ hai)

Hum do aisi numbers dhundhte hain jo:

Multiply hokar $c$ den (constant term)
Add hokar $b$ den ( $x$ ka coefficient)

Kyun? Kyunki $(x + p) (x + q) = x^{2} + (p + q) x + pq$ . Coefficients match karne par:

$x$ ka coefficient: $p + q = b$
Constant term: $pq = c$

Hum kya karte hain: $c$ ke factor pairs dhundho, test karo ki kaunsa pair $b$ mein add hota hai.

Yeh step kyun? Har factor ek "boundary" represent karta hai jahan parabola x-axis ko cross karti hai. Har factor ko zero set karne se woh crossing points isolate ho jaate hain.

Case 2: Non-monic quadratics ( $a \neq = 1$ , toh hamare paas $a x^{2} + b x + c = 0$ hai)

AC Method (Derivation):

$a \cdot c$ multiply karo ek target product paane ke liye.
Do aisi numbers dhundho jo $a c$ mein multiply hon aur $b$ mein add hon.
Middle term $b x$ ko un do numbers se split karo.
Grouping se factor karo.

Yeh kaam kyun karta hai? Hum problem ko monic-jaisi structure mein transform kar rahe hain, leading coefficient ko temporarily constant term mein absorb karke.

Yeh step kyun? Grouping ek common factor ko isolate karta hai, hidden factorization reveal karta hai.

Figure — Quadratic equations — factoring, completing the square

Method 2: Completing the Square

First Principles se Derivation

Goal: $a x^{2} + b x + c$ ko aise rewrite karo ki $x^{2}$ aur $x$ terms $(x + kuch)^{2}$ mein combine ho jaayein.

Step-by-step construction:

$a x^{2} + b x + c = 0$ se shuru karo.

Step 1: Pehle do terms se $a$ factor out karo (andar $x^{2}$ coefficient 1 banao): $a (x^{2} + \frac{b}{a} x) + c = 0$

Kyun? Ek perfect square $(x + k)^{2} = x^{2} + 2 k x + k^{2}$ ke liye $x^{2}$ coefficient 1 hona zaroori hai.

Step 2: Square complete karne ke liye number identify karo. Ek perfect square $(x + k)^{2}$ ka middle term $2 k$ hota hai. Toh: $2 k = \frac{b}{a} \Rightarrow k = \frac{b}{2 a}$

Perfect square mein missing constant term hai $k^{2} = (\frac{b}{2 a})^{2} = \frac{b ^{2}}{4 a ^{2}}$ .

Step 3: Yeh term parentheses ke andar add aur subtract karo: $a (x^{2} + \frac{b}{a} x + \frac{b ^{2}}{4 a ^{2}} - \frac{b ^{2}}{4 a ^{2}}) + c = 0$

Add aur subtract kyun? Hum equation ki value change nahi kar rahe, bas usse rearrange kar rahe hain.

Step 4: Perfect square pehchano aur simplify karo: $a [(x + \frac{b}{2 a})^{2} - \frac{b ^{2}}{4 a ^{2}}] + c = 0$ $a (x + \frac{b}{2 a})^{2} - \frac{b ^{2}}{4 a} + c = 0$ $a (x + \frac{b}{2 a})^{2} = \frac{b ^{2}}{4 a} - c = \frac{b ^{2} - 4 a c}{4 a}$

Step 5: Dono sides ko $a$ se divide karo: $(x + \frac{b}{2 a})^{2} = \frac{b ^{2} - 4 a c}{4 a ^{2}}$

Step 6: Dono sides ka square root lo: $x + \frac{b}{2 a} = \pm \frac{b ^{2} - 4 a c}{2 a}$

Step 7: $x$ ke liye solve karo: $x = - \frac{b}{2 a} \pm \frac{b ^{2} - 4 a c}{2 a} = \frac{- b \pm b ^{2} - 4 a c}{2 a}$

Verification: Wapas substitute karo: $(- 1)^{2} + 6 (- 1) + 5 = 1 - 6 + 5 = 0$ ✓, aur $(- 5)^{2} + 6 (- 5) + 5 = 25 - 30 + 5 = 0$ ✓.

Mistake 1: Pehle $a$ factor out karna bhool jaana Student likhta hai: $2 x^{2} + 8 x + 6 = 2 (x + 2)^{2}$ ✗

Yeh sahi kyun lagta hai: Student 2 factor out karta hai, $2 (x^{2} + 4 x + 3)$ dekhta hai, 4 ka aadha lekar 2 paata hai, aur seedha $2 (x + 2)^{2}$ par pahunch jaata hai—yeh bhoolte hue ki $(x + 2)^{2} = x^{2} + 4 x + 4$ ek extra $+ 4$ introduce karta hai jiske liye compensate karna padta hai.

Fix: Pehle hamesha $x^{2}$ coefficient ko 1 ke barabar banao, phir completing term add aur subtract karo: $2 (x^{2} + 4 x + 3) = 2 [(x + 2)^{2} - 4 + 3] = 2 [(x + 2)^{2} - 1]$ . Bacha hua $- 1$ matter karta hai!

Mistake 2: Factoring mein sign errors Student $x^{2} - 5 x + 6$ ko $(x - 2) (x - 3)$ factor karta hai lekin phir solutions $x = - 2, x = - 3$ likhta hai ✗

Yeh sahi kyun lagta hai: Factors mein $- 2$ aur $- 3$ hain, toh student sochta hai ki wahi directly answers hain.

Fix: Yaad rakho, hum har factor ko zero ke barabar set karte hain: $x - 2 = 0 \Rightarrow x = + 2$ . Solve karte waqt signs flip ho jaate hain. Poochho: "Kaunsi value har factor ko zero banati hai?"

Mistake 3: Completing term sirf ek side mein add karna Student likhta hai: $x^{2} + 6 x = - 5$ , phir $x^{2} + 6 x + 9 = - 5$ ✗

Yeh sahi kyun lagta hai: Woh correctly completing-the-square term left mein add karte hain lekin bhool jaate hain ki equation balanced rehni chahiye.

Fix: Left mein jo add karo, right mein bhi add karo. $x^{2} + 6 x + 9 = - 5 + 9 = 4$ .

Recall Ek 12-Saal-Ke Bachche Ko Explain Karo

Socho tumhare paas ek tedha bridge hai (ek parabola) aur tum jaanna chahte ho ki woh zameen ko kahan touch karta hai. Wahi jagah hai jahan height zero ke barabar hoti hai.

Factoring ek badi LEGO structure ko do chhotey towers mein todne jaisa hai. Agar poori cheez zameen par flat hai (zero ke barabar hai), toh ek tower flat hona chahiye (ek factor zero ke barabar hona chahiye). Hum dhundhhte hain ki kaunse pieces har tower ko collapse karte hain.

Completing the square puzzle pieces ko rearrange karne jaisa hai ek perfect square picture banane ke liye. Jab ek baar square dikh jaata hai, toh aasani se measure kar sakte ho ki center se edges kitni door hain—aur woh distances batati hain ki bridge zameen ko kahan touch karta hai. Yeh ek clever trick hai jo har baar kaam karti hai, tab bhi jab factoring bahut mushkil ho!

Dono methods tumhari math toolbox mein tools hain. Factoring tab fast hoti hai jab tum pattern spot kar sako. Completing the square tumhari superpower hai jab numbers messy ho jaayein—isi tarah humne quadratic formula discover kiya tha!

Factoring (AC method): " $a c$ Multiply karo, Find two numbers, Middle term Split karo, Group karo, Factor out karo"

Connections

2.1.15-Linear-equations-and-inequalities — Factoring quadratics ko linear equations mein reduce karta hai
2.1.17-Quadratic-formuland-discriminant — Completing the square quadratic formula derive karta hai
2.3.4-Parabolas-and-vertex-form — Completing the square vertex form $y = a (x - h)^{2} + k$ reveal karta hai
3.2.1-Polynomials-factoring-techniques — Factoring higher-degree polynomials tak extend hoti hai
4.1.2-Solving-systems-graphically — Quadratic solutions graph par x-intercepts hote hain

Flashcards

Quadratic equation kya hoti hai? :: Ek aisi equation jiska form $a x^{2} + b x + c = 0$ ho jahan $a \neq = 0$ .

Zero Product Property kya hai?

Agar

A \cdot B = 0

, toh

A = 0

B = 0

(ya dono).

x^{2} + b x + c

factor karte waqt, factor pair ko kaunse do properties satisfy karni chahiye?

Unhe

c

mein multiply hona chahiye aur

b

mein add hona chahiye.

AC method mein "target product" kya hota hai?

a \cdot c

(leading coefficient aur constant term ka product).

a x^{2} + b x + c = 0

ke liye completing the square ka pehla step kya hai?

x^{2}

aur

x

terms se

a

factor out karo taaki

x^{2}

coefficient andar 1 ho jaaye.

x^{2} + b x

ke liye completing the square karte waqt hum kaunsa term add (aur subtract) karte hain?

(\frac{b}{2})^{2}

, jo

x

-coefficient ke aadhe ka square hai.

Quadratic equation mein discriminant kya hota hai?

Δ = b^{2} - 4 a c

, quadratic formula mein square root ke neeche wala expression.

Completing the square ke baad,

2 x^{2} + 8 x + 3 = 0

ka form

2 (x + 2)^{2} = k

ban jaata hai.

k

kya hai?

2 (x + 2)^{2} = 5

, toh

k = 5

(x - p) (x - q) = 0

solve karte waqt har factor ko zero ke barabar kyun set karte hain?

Kyunki ek product zero ke barabar tab hi hota hai jab kam se kam ek factor zero ho (Zero Product Property).

Completing the square kaunsi form produce karta hai?

(x - h)^{2} = k

ya parabola ke liye vertex form

a (x - h)^{2} + k