find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the coefficient
xsq

2 thoughts on “find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the coefficient <br />xsq”

1. Answer:

   Correct Question :-

   • Find the zeroes of the following quadratic equation polynomial x² – 5 and verify the relationship between the zeroes and co-efficient.

   Given :-

   • x² – 5

   To Find :-

   • What is the zeroes of the quadratic equation.
   • Verify the relationship between the zeroes and co-efficient.

   Formula Used :-

   [tex]\clubsuit[/tex] Sum of roots :

   [tex]\longmapsto \: \sf\boxed{\bold{\pink{Sum\: of\: roots\: (\alpha + \beta) =\: \dfrac{- b}{a}}}}[/tex]

   [tex]\clubsuit[/tex] Product of roots :

   [tex]\longmapsto \sf \boxed{\bold{\pink{Product\: of\: roots\: (\alpha\beta) =\: \dfrac{c}{a}}}}[/tex]

   Solution :-

   Given :

   [tex]\dashrightarrow \sf\bold{\green{x^2 – 5}}[/tex]

   [tex]\implies \sf p(x) =\: x^2 – 5[/tex]

   [tex]\implies \sf x^2 – 5 =\: 0[/tex]

   [tex]\implies \sf x^2 =\: 5[/tex]

   [tex]\implies \sf x =\: \sqrt{5}[/tex]

   [tex]\implies \sf\bold{\red{x =\: \sqrt{5}}}[/tex]

   And,

   [tex]\implies \sf\bold{\red{x =\: – \sqrt{5}}}[/tex]

   [tex]\therefore[/tex] The zeroes of quadratic polynomial is √5 and – √5.

   Hence,

   • α = √5
   • β = – √5

   [tex]\rule{150}{2}[/tex]

   [tex]\bigstar[/tex] Verify the relationship between the zeroes and co-efficient :

   Given equation :

   [tex]\dashrightarrow \sf x^2 – 5[/tex]

   where,

   • a = 1
   • b = 0
   • c = – 5

   [tex]\leadsto \sf\bold{Sum\: of\: roots\: :-}[/tex]

   [tex]\implies \sf \sqrt{5} + (- \sqrt{5}) =\: \dfrac{- 0}{1}[/tex]

   [tex]\implies \sf {\cancel{\sqrt{5}}} – {\cancel{\sqrt{5}}} =\: 0[/tex]

   [tex]\implies \sf\bold{\red{0 =\: 0}}[/tex]

   [tex]\leadsto \sf\bold{Product\: of\: roots}[/tex]

   [tex]\implies \sf \sqrt{5} \times (- \sqrt{5}) =\: \dfrac{- 5}{1}[/tex]

   [tex]\implies \sf – \sqrt{25} =\: – 5[/tex]

   [tex]\implies \sf\bold{\red{- 5 =\: – 5}}[/tex]

   Hence, Verified.

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2. ✯✯ QUESTION ✯✯

   Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and coefficients of x²-9..

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   ✰✰ ANSWER ✰✰

   [tex]\longrightarrow{{x}^{2}-9} [/tex]

   Using Identity : –

   [tex]\longrightarrow{{a}^{2}-{b}^{2}=(a+b)(a-b)}[/tex]

   [tex]\longrightarrow{({x}^{2}+(-9)({x}^{2}-(-9)}[/tex]

   [tex]\longrightarrow{({x}^{2}-9)({x}^{2}+9)}[/tex]

   Now ,

   [tex]\longrightarrow{{x}^{2}-9=0}[/tex]

   [tex]\longrightarrow{{x}^{2}=\sqrt{9}} [/tex]

   [tex]\red\longmapsto\:\large\underline{\boxed{\bf\green{x}\orange{=}\purple{3}}}[/tex]

   [tex]\longrightarrow{{x}^{2}+9=0} [/tex]

   [tex]\longrightarrow{{x}^{2}=\sqrt[-]{9}} [/tex]

   [tex]\red\longmapsto\:\large\underline{\boxed{\bf\orange{x}\green{=}\blue{-3}}}[/tex]

   So , 3 and -3 are the zeroes of polynomials x²-9..

   Now ,

   ➥Sum of Zeroes : –

   [tex]\longrightarrow{\alpha+\beta=\dfrac{-b}{a}}[/tex]

   [tex]\longrightarrow{3+(-3)=\dfrac{-(-0)}{1}}⟶3+(−3)[/tex]

   [tex]\longrightarrow{0=0}[/tex]

   ➥Product of Zeroes : –

   [tex]\longrightarrow{\alpha\beta=\dfrac{c}{a}} [/tex]

   [tex]\longrightarrow{3\times{-3}=\dfrac{-9}{1}} [/tex]

   [tex]\longrightarrow{-9=-9}[/tex]

   [tex]\pink\longmapsto\:\large\underline{\boxed{\bf\purple{L.H.S}\green{=}\red{R.H.S}}}[/tex]

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