Question

If 2 and -3 are the zeroes of the quadratic polynomial x²+(a+1)x+b, then find the values of a and b.If 2 and -3 are the zeroes of the quadratic polynomial x²+(a+1)x+b, then find the values of a and b.​

Answer 1

GivEn :

• 2 & -3 are Zeroes of Polynomial x²+(a+1)x+b .

To Find :

• Value of a & b ?

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⠀

⠀⠀⠀⠀⠀⠀⠀⠀✠ G I V E N⠀P O L Y N O M I A L : x² + ( a + 1 ) x + b

⌑ By Comparing the given polynomial with Standard Quadratic equation (ax² + bx + c = 0), we get —

• a = 1 ,
• b = ( a + 1 ) &
• c = b

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⠀

⠀✇ Now , Finding Value of a and b :

$\dag\frak{\underline { As, \:We \:know \:that\::\:}}$

• { α + β } = – b/ a
• { α β } = c/a

$\qquad \qquad\qquad \dag\underline {\: \pink{\pmb {\:\sf Sum \: of \:roots \:\:\big\{ \alpha + \beta \:\big\} \:\:}}}$

$\qquad \qquad \twoheadrightarrow \tt { \:\:\big\{ \alpha + \beta \:\big\} \:=\:\dfrac{-\:b}{a}\:}$

$\qquad \qquad \twoheadrightarrow \tt { \:\:\big\{ 2 + ( -3 ) \:\big\} \:=\:\dfrac{-\:( a + 1 )}{1}\:}$

$\qquad \qquad \twoheadrightarrow \tt { \:\:\big\{ 2 - 3 \:\big\} \:=\:\dfrac{- a - 1 )}{1}\:}$

$\qquad \qquad \twoheadrightarrow \tt { \:\:\big\{ 2 - 3 \:\big\} \:=\:\dfrac{- a - 1 )}{1}\:}$

$\qquad \qquad \twoheadrightarrow \tt { \:\: -1 \:=\:\dfrac{- a - 1 )}{1}\:}$

$\qquad \qquad \twoheadrightarrow \tt { \:\: -a \:=\:\dfrac{ 1 - 1 )}{1}\:}$

$\qquad \qquad \twoheadrightarrow \underline {\pmb{\boxed {\tt { \purple { \:\: a \:=\:0\:}}}}}\:\:\bigstar$

$\qquad \qquad\qquad \dag\underline {\: \pink{\pmb {\:\sf Product \: of \:roots \:\:\big\{ \alpha \beta \:\big\} \:\:}}}$

$\qquad \qquad \twoheadrightarrow \tt { \:\:\big\{ \alpha \beta \:\big\} \:=\:\dfrac{\:c}{a}\:}$

$\qquad \qquad \twoheadrightarrow \tt { \:\:\big\{ 2 \:\times ( -3 ) \:\big\} \:=\:\dfrac{\:b}{1}\:}$

$\qquad \qquad \twoheadrightarrow \tt { \:\: -6 \:=\:\:b\:}$

$\qquad \qquad \twoheadrightarrow\underline{ \pmb{\boxed {\tt { \purple { \:\: b \:=\:-6\:}}}}}\:\:\bigstar$

$\qquad \therefore \:\:\underline {\sf \:Hence \:Values \:of \:a \: and \:b \:are \:\pmb{\sf 0 }\:and \:\pmb{ \sf -6 }\:,respectively \:.}$