2.1.20
Agar aapko pata hai ki ek parabola x-axis ko kahan touch karti hai (uske roots), toh aap poori equation ko reverse-engineer kar sakte ho. Roots ko "DNA" samjho — yeh quadratic ki structure ko uniquely determine karte hain. Jab x = α ( x − α ) α β ( x − α ) ( x − β )
Ek quadratic equation ke roots α β
Step 1: "Root" ka matlab kya hai?
Agar α P ( x ) P ( α ) = 0 ( x − α )
Step 2: Do roots matlab do factors
Agar dono α β
( x − α ) P ( α ) = 0 ( x − β ) P ( β ) = 0
Step 3: Factors ko multiply karo
Dono factors wala sabse simple polynomial unka product hai:
k kahan cross karti hai, yeh nahi ki woh kitni stretched hai. Ye saari parabolas ke roots same hain lekin "width" alag hai:
k = 1 k = 2 k = 0.5
Coefficients dekho! Agar a x 2 + b x + c = 0
Yeh kyun important hai: Aap sirf sum aur product se quadratic bana sakte ho, bina individual roots jaane!
x 2 − ( sum of roots ) x + ( product of roots ) = 0
Worked example Example 1: Individual Roots Diye HainProblem: Roots α = 3 β = − 5 Solution:
Step 1: Factor form use karo
P ( x ) = k ( x − 3 ) ( x − ( − 5 )) = k ( x − 3 ) ( x + 5 )
Yeh step kyun? Har root ek aisa factor contribute karta hai jo us root par zero ho jaata hai.
Step 2: Expand karo (sabse simple form ke liye k = 1 P ( x ) = ( x − 3 ) ( x + 5 ) = x 2 + 5 x − 3 x − 15 = x 2 + 2 x − 15 Expand kyun karte hain? Standard form a x 2 + b x + c
Answer: x 2 + 2 x − 15 = 0
Verify karo: Sum = 3 + ( − 5 ) = − 2 = − 1 2 = 3 × ( − 5 ) = − 15 = 1 − 15
Worked example Example 2: Sum aur Product Diye HainProblem: Ek quadratic banao jiske roots ka sum = 7 = 10
Solution:
Direct formula:
x 2 − ( sum ) x + ( product ) = 0 x 2 − 7 x + 10 = 0 Yeh kyun kaam karta hai? Vieta's formulas se: α + β = 7 α β = 10 ( x − α ) ( x − β )
Actual roots dhundhna (bonus check):
Factor karo: ( x − 5 ) ( x − 2 ) = 0 α = 5 , β = 2 5 + 2 = 7 5 × 2 = 10
Worked example Example 3: Radicals Wale RootsProblem: Roots 2 + 3 2 − 3
Solution:
Method 1 (factor form):
P ( x ) = [ x − ( 2 + 3 )] [ x − ( 2 − 3 )] Difference of squares pattern se expand karo:
Maano a = x − 2 P ( x ) = ( a − 3 ) ( a + 3 ) = a 2 − 3 = ( x − 2 ) 2 − 3 = x 2 − 4 x + 4 − 3 = x 2 − 4 x + 1
Yeh step kyun? Conjugate pairs ( a − b ) ( a + b ) = a 2 − b 2
Method 2 (sum aur product):
Sum: ( 2 + 3 ) + ( 2 − 3 ) = 4
Product: ( 2 + 3 ) ( 2 − 3 ) = 4 − 3 = 1
Equation: x 2 − 4 x + 1 = 0
Answer: x 2 − 4 x + 1 = 0
Worked example Example 4: Scaled QuadraticProblem: Roots − 2 3 ( 1 , 8 )
Solution:
Step 1: Roots ke saath generic form
P ( x ) = k ( x + 2 ) ( x − 3 ) Step 2: Point condition P ( 1 ) = 8 8 = k ( 1 + 2 ) ( 1 − 3 ) 8 = k ( 3 ) ( − 2 ) 8 = − 6 k k = − 3 4
Yeh step kyun? k k
Step 3: Final equation likho
P ( x ) = − 3 4 ( x + 2 ) ( x − 3 ) = − 3 4 ( x 2 − x − 6 ) = − 3 4 x 2 + 3 4 x + 8
Ya standard form mein: 4 x 2 − 4 x − 24 = 0 − 3
Common mistake Mistake 1: Factors mein Sign ErrorsGalat: Roots 3 − 5 ( x + 3 ) ( x − 5 ) = 0 Kyun sahi lagta hai: "Root 3 3
Fix: Agar α ( x − α ) ( x + α )
Root = 3 = ( x − 3 ) x = 3 = 0
Root = − 5 = ( x − ( − 5 )) = ( x + 5 ) x = − 5 = 0
Sahi: ( x − 3 ) ( x + 5 ) = 0
Common mistake Mistake 2:k Galat: "Roots 2 5 x 2 − 7 x + 10 = 0 Kyun sahi lagta hai: Hum diye gaye roots ke liye "the" equation seekhte hain, jo uniqueness imply karta hai.
Fix: Infinite quadratics same roots share kar sakte hain! Woh ek constant multiple se alag hote hain:
x 2 − 7 x + 10 = 0 2 x 2 − 14 x + 20 = 0 − x 2 + 7 x − 10 = 0 2 5 k ( x 2 − 7 x + 10 ) = 0 k = 0
Common mistake Mistake 3: Sum/Product ka Galat FormulaGalat: Roots ka sum 5 6 x 2 + 5 x + 6 = 0
Kyun sahi lagta hai: Diye gaye values ko directly use karna.
Fix: Formula hai x 2 − ( sum ) x + ( product ) = 0 ( x − α ) ( x − β ) = x 2 − ( α + β ) x + α β Sahi: x 2 − 5 x + 6 = 0 2 3 = 5 = 6
Recall Feynman Explanation (12 saal ke bachche ko samjhao)Socho tum ek treasure hunt game khel rahe ho. Tumhe do treasure chests milte hain jo number line par positions 3 5 rule de sakte ho jo bataye ki treasures kahan hain?"
Tum keh sakte ho: "Koi bhi position x ( x − 3 ) ( x − 5 ) ( x − 3 ) × ( x − 5 )
Yahan magic hai: yeh multiplication exactly ZERO hota hai jab x
x = 3 ( 3 − 3 ) × ( 3 − 5 ) = 0 × ( − 2 ) = 0 x = 5 ( 5 − 3 ) × ( 5 − 5 ) = 2 × 0 = 0 Kahi aur: dono factors non-zero hain, toh product non-zero hai
Jab tum ( x − 3 ) ( x − 5 ) x 2 − 8 x + 15 = 0
Mnemonic Memory Aid: "ROOT ko SUBTRACT karo"S ubtract the R oot to form the factor.
Root hai + 7 ( x − 7 ) − 3 ( x − ( − 3 )) = ( x + 3 ) Sum/product formula ke liye: "S um S ubtract hota hai"
x 2 − ( sum ) x + ( product ) = 0
Related concepts:
Vieta's formulas — direct link, yahan se hi sum/product relationships aate hainFactor theorem — theoretical basis ke liye ki ( x − α ) Quadratic formula — inverse operation: roots → equation vs equation → rootsCompleting the square — quadratic structure manipulate karne ka alternative tarikaPolynomial long division — higher-degree polynomials ke roots se equations dhundhne tak extend hota haiComplex roots — jab α β Graph transformations — k
Yeh kab use hoga:
Optimization problems: Critical points (derivative ke roots) dhundhne ke baad, original function reconstruct karoCurve fitting: Jab data points diye hon jahan parabola x-axis cross karti haiEngineering: Specified landing points wale projectile paths design karoPhysics: Known equilibrium points wale harmonic oscillators model karo
#flashcards/maths
What is the general form of a quadratic with roots α and β? :: k ( x − α ) ( x − β ) k = 0 k [ x 2 − ( α + β ) x + α β ]
If roots of a quadratic are 4 and -3, what is the equation (with k=1)? ( x − 4 ) ( x + 3 ) = x 2 − x − 12 = 0
What is the formula for a quadratic given sum S and product P of roots? x 2 − S x + P = 0
Why does the factor form ( x − α ) ( x − β ) At x = α ( x − α ) = 0 0 x = β ( x − β ) = 0 0
If sum of roots is 10 and product is 21, form the equation :: x 2 − 10 x + 21 = 0
What does the constant k in k ( x − α ) ( x − β ) Parabola ki vertical scaling — determine karta hai ki woh kitni "stretched" ya "compressed" hai, lekin root locations nahi badalta
If roots are 3 + 5 3 − 5 Sum =
6 , Product =
( 3 ) 2 − ( 5 ) 2 = 9 − 5 = 4 . Equation:
x 2 − 6 x + 4 = 0 Common mistake: If root is -7, what is the factor? ( x − ( − 7 )) = ( x + 7 ) ( x − 7 )
From Vieta's formulas, if 2 x 2 − 8 x + 6 = 0 a c = 2 6 = 3
Why are there infinitely many quadratics with the same two roots? Koi bhi non-zero multiple k ek alag quadratic deta hai: k ( x − α ) ( x − β )
Root alpha: P of alpha = 0
x-alpha and x-beta factors
P = k times x-alpha times x-beta
Form quadratic from sum and product
x^2 - sum x + product = 0
