Find the quadratic polynomial in each case (i) sum and product of the zeroes respectively
are _1/4 and 1/4
and (ii) for the zeroes a, ß are 2, – 1.
Find the quadratic polynomial in each case (i) sum and product of the zeroes respectively
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▪IN THE FIRST QUESTION :
☯GIVEN :
☯TO FIND :
☯FORMULA REQUIRED :
➲SOLUTION :
[tex] :\implies {\sf x^2 – \left(\alpha + \beta \right)x + \alpha \beta = 0}\\ \\ [/tex]
[tex]:\implies {\sf x^2- (\dfrac{-1}{4} )x+\dfrac{1}{4} =0 }\\ \\ [/tex]
[tex]:\implies {\sf x^2+ \dfrac{1}{4}x+\dfrac{1}{4} =0 }\\ \\ [/tex]
[tex]:\implies {\sf \dfrac{4x^2 +x + 1}{4} =0 }\\ \\[/tex]
[tex]:\implies {\sf 4x^2+x + 1 = 0 \times 4} \\ \\ [/tex]
[tex]:\implies{ \underline{ \boxed{ \blue{\bf{ 4 x^2 +x + 1 = 0}}}}}[/tex]
[tex]\huge{\green{\therefore}}[/tex]Quadratic Polynomial = [tex]\bf{ 4x^2 + x + 1 }[/tex]
▪IN THE SECOND QUESTION :
☯GIVEN :
☯TO FIND :
☯FORMULA REQUIRED :
➲SOLUTION :
★ Sum of Zeros :
[tex]: \leadsto \alpha + \beta \\ \\[/tex]
[tex]: \leadsto2 + (-1) \\ \\ [/tex]
[tex]: \leadsto 2-1 \\ \\ [/tex]
[tex]: \leadsto \bf{1}[/tex]
★ Product of Zeros :
[tex] : \leadsto \alpha \beta \\ \\ [/tex]
[tex]: \leadsto 2 \times (-1)\\ \\[/tex]
[tex]: \leadsto \bf{ -2 }[/tex]
[tex]:\implies {\sf x^2 – \left(\alpha + \beta \right)x + \alpha \beta = 0}\\ \\ [/tex]
[tex]:\implies {\sf x^2- \left( 1\right )x+(-2) =0 }\\ \\ [/tex]
[tex]: \implies \underline{\boxed{ \purple{\bf{ x ^2 – x – 2 = 0 }}}}[/tex]
[tex]\huge{\green{\therefore}}[/tex] Quadratic polynomial = [tex]\bf{ x ^2 – x -2}[/tex]