Question

Find the quadratic polynomial with zeroes 2 and

−1 /3

​

Answer 1

☯GIVEN :

• $\bf{\alpha = 2 }$

• $\bf{\beta = \dfrac{ - 1}{3} }$

☯TO FIND :

• Quadratic Polynomial.

☯FORMULA REQUIRED :

• $\underline{\boxed{\purple{\bf{x^2- \left(\alpha + \beta\right)x + \alpha\beta =0 }}}}$

➲SOLUTION :

★ Sum of Zeros :

$: \leadsto \alpha + \beta \\ \\ : \leadsto2 + \dfrac{ - 1}{3}$

$: \leadsto\dfrac{ 6 + (- 1)}{3}$

$: \leadsto\dfrac{ 6 - 1}{3}$

$: \leadsto \bf{\dfrac{ 5}{3}}$

★ Product of Zeros :

$: \leadsto \alpha \beta$

$: \leadsto 2 \times \dfrac{ - 1}{3}$

$: \leadsto \bf{ \dfrac{ - 2}{3} }$

• By substituting the values, $\alpha + \beta = \dfrac{ 5}{3}$ and $\alpha \beta = \dfrac{-2}{3}$

$:\implies {\sf x^2 - \left(\alpha + \beta \right)x + \alpha \beta = 0}$

$:\implies {\sf x^2- \left( \dfrac{5}{3} \right )x+\dfrac{ - 2}{3} =0 }$

$:\implies {\sf x^2- \dfrac{5}{3} x - \dfrac{ 2}{3} =0 }$

$: \implies \sf{ \dfrac{ 3x ^2 - 5x - 2}{3} = 0}$

• By cross multiplication :

$: \implies \sf{ 3x ^2 - 5x - 2 = 0 \times 3}$

$: \implies \underline{\boxed{ \purple{\bf{ 3x ^2 - 5x - 2 = 0 }}}}$

$\huge{\green{\therefore}}$Quadratic polynomial = $\bf{ 3x ^2 - 5x -2}$