Question

If a and b are the zeroes of the quadratic polynomial f(x) = x2 – 2x + 3, find a polynomial whose roots are a+2, b+2.

Answer 1

$\large{\underline{\bf{\purple{Given:-}}}}$

✦ f(x) = x² - 2x + 3

$\large{\underline{\bf{\purple{To\:Find:-}}}}$

✦ we need to find a polynomial whose roots are a +2 and b + 2 .

$\huge{\underline{\bf{\red{Solution:-}}}}$

f(x) = x² - 2x + 3

Now ,

• a = 1
• b = -2
• c = 3

sum of zeroes = - b/a

• α + β = 2

product of zeroes = c/a

• αβ = 3

Now , quadratic polynomial:-

$\leadsto \rm\:\:$x² -[(α + 2)+(β+2)]x + (α+2)(β+2)

$\leadsto \rm\:\:$x² -(α + β +4)x + αβ + 2(α + β) + 4

$\leadsto \rm\:\:$x² -(2 +4)x +3 +2(2) +4

$\leadsto \rm\:\:$x² - 6x + 11

So,

The quadratic polynomial is x² - 6x + 11 whose zeroes are a+2 and b+2 .

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Answer 2

$\bold\red{\underline{\underline{Answer:}}}$

$\bold{The \ required \ quadratic \ polynomial \ is}$

$\bold{x^{2}-6x+11.}$

$\bold\orange{Given:}$

$\bold{The \ given \ quadratic \ polynomial \ is}$

$\bold{=>x^{2}-2x+3}$

$\bold{=>It's \ zeroes \ are \ a \ and \ b}$

$\bold\pink{To \ find:}$

$\bold{A \ polynomial \ whose \ roots \ are }$

$\bold{a+2 \ and \ b+2.}$

$\bold\green{\underline{\underline{Solution}}}$

$\bold{The \ given \ quadratic \ polynomial \ is}$

$\bold{=>x^{2}-2x+3}$

$\bold{Here, \ a=1, \ b=-2 \ and \ c=3}$

$\bold{Sum \ of \ roots=\frac{-b}{a}}$

$\bold{\therefore{a+b=2...(1)}}$

$\bold{Product \ of \ roots=\frac{c}{a}}$

$\bold{\therefore{a×b=3...(2)}}$

________________________________

$\bold{Let \ \alpha \ and \ \beta \ be \ roots}$

$\bold{of \ new \ quadratic \ polynomial.}$

$\bold{=>\alpha=a+2}$

$\bold{=>\beta=b+2}$

$\bold{\alpha+\beta=a+2+b+2}$

$\bold{\therefore{\alpha+\beta=4+a+b}}$

$\bold{from \ (1)}$

$\bold{\alpha+\beta=4+2}$

$\bold{\therefore{\alpha+\beta=6...(3)}}$

$\bold{Product \ of \ roots=\frac{c}{a}}$

$\bold{\therefore{\alpha×\beta=(a+2)(b+2)}}$

$\bold{\therefore{\alpha×\beta=ab+2(a+b)+4}}$

$\bold{from \ (1) \ and (2)}$

$\bold{\alpha×\beta=3+2(2)+4}$

$\bold{\alpha×beta=11...(4)}$

$\bold{Quadratic \ polynomial \ is}$

$\bold{=>x^{2}-(sum \ of \ roots)x+(product \ of \ roots)}$

$\bold{from \ (3) \ and \ (4)}$

$\bold{=>x^{2}-6x+11}$

$\bold\purple{\tt{\therefore{The \ required \ quadratic \ polynomial \ is}}}$

$\bold\purple{\tt{x^{2}-6x+11.}}$