Question

Find the zeroes of quaderatic polynomial and verify relationship between the zeroes and the ciefficients.2x^2+5x+3?​

Answer 1

SOLUTION:-

Given

• 2x²+5x+3=0

To find

• Zeros

• and verify relationship b/w Zeros and coefficients

Explanation

☆__ 2x²+5x+3=0__ ☆

➡️ 2x²+(2+3)x+3=0

➡️ 2x²+ 2x+3x+3=0

➡️ 2x(x+1) +3( x+1)=0

➡️(x+1)(2x+3)=0

➡️x+1=0, 2x+3=0

➡️ x=-1 , x= -3/2

Zeros are ==>>> -1 and -3/2

Sum of zeros = -b/a

➡️ -1+(-3/2) = -5/2

➡️ -1/1 - 3/2 = -5/2

➡️ -2-3 / 2 = -5/2

➡️ -5/2 = -5/2

Product of zeros = c/a

➡️ -1×(-3/2) = 3/2

➡️ 3/2 = 3/2

Hence verified !!

_______________________________

Answer 2

Solution :

We have quadratic polynomial p(x) = 2x² + 5x + 3 & zero of the polynomial p(x) = 0

Now, we factorized;

$\longrightarrow\sf{2x^{2} + 5x + 3=0}\\\\\longrightarrow\sf{2x^{2} +2x + 3x + 3=0}\\\\\longrightarrow\sf{2x(x + 1) + 3(x+1)=0}\\\\\longrightarrow\sf{(x+1)(2x+3)=0}\\\\\longrightarrow\sf{x+1=0\:\:Or\:\:2x+3=0}\\\\\longrightarrow\sf{x=-1\:\:Or\:\:2x=-3}\\\\\longrightarrow\bf{x=-1\:\:Or\:\:x=-3/2}$

∴ α = -1 & β = -3/2 are the zeroes of the given polynomial.

As we know that given polynomial compared with ax² + bx + c;

• a = 2
• b = 5
• c = 3

Now;

$\underline{\mathcal{SUM\:OF\:THE\:ZEROES\::}}$

$\mapsto\sf{\alpha +\beta =\dfrac{-b}{a} =\bigg\lgroup\dfrac{Coefficient\:of\:x}{Coefficient\:of\:x^{2}} \bigg\rgroup }\\\\\\\mapsto\sf{-1+\bigg(-\dfrac{3}{2} \bigg) = \dfrac{-5}{2} }\\\\\\\mapsto\sf{-1-\dfrac{3}{2} = \dfrac{-5}{2} }\\\\\\\mapsto\sf{\dfrac{-2-3}{2} = \dfrac{-5}{2} }\\\\\\\mapsto\bf{\dfrac{-5}{2} = \dfrac{-5}{2} }$

$\underline{\mathcal{PRODUCT\:OF\:THE\:ZEROES\::}}$

$\mapsto\sf{\alpha \times \beta =\dfrac{c}{a} =\bigg\lgroup\dfrac{Constant\:term}{Coefficient\:of\:x^{2}} \bigg\rgroup }\\\\\\\mapsto\sf{-1\times \bigg(-\dfrac{3}{2} \bigg) = \dfrac{3}{2} }\\\\\\\mapsto\sf{-\bigg(-\dfrac{3}{2}\bigg) = \dfrac{3}{2} }\\\\\\\mapsto\bf{\dfrac{3}{2} = \dfrac{3}{2} }$

Thus;

The relations between zeroes & coefficient are verified .