Question

find the zero of quadratic polynomial 4 x square - 9 and verify the relationship between the zeros and its composition​

Answer 1

Let p(x)=4x^2-9

To find zeroes make p(x)=0

4x^2-9=0

(2x)^2-3^2=0

(2x+3)(2x-3)=0

Therefore

2x+3=0 or 2x-3=0

2x=-3 or 2x=3

x=-3/2 or x=3/2

Two zeroes are p=-3/2, q=3/2

ii) compare given equation with ax^2+bx+c=0

a=4, b=0,c=-9

Sum of the zeroes = -b/a

p+q= -0/4=

p+q=-3/2+3/2=0

iii)product of the zeroes =c/a

pq=-9/4

(-3/2)(3/2)=-9/4

Answer 2

Solution :

$\bf{\red{\underline{\bf{Given\::}}}}$

The quadratic polynomial 4x² - 9.

$\bf{\red{\underline{\bf{To\:find\::}}}}$

The zeroes and verify the relationship between zero and it's coefficient.

$\bf{\red{\underline{\bf{Explanation\::}}}}$

We have p(x) = 4x² - 9

Zero of the polynomial is p(x) = 0

So;

$\longrightarrow\tt{4x^{2} -9=0}\\\\\longrightarrow\tt{(2x)^{2} -(3)^{2} =0}\\\\\longrightarrow\tt{(2x+3)(2x-3)=0\:\:\:[\therefore using \:a^{2} -b^{2} ]}\\\\\longrightarrow\tt{2x+3=0\:\:\:Or\:\:\:2x-3=0}\\\\\longrightarrow\tt{2x=-3\:\:\:Or\:\:\:2x=3}\\\\\longrightarrow\tt{\pink{x=\dfrac{-3}{2} \:\:\:Or\:\:\:x=\dfrac{3}{2} }}$

∴ The α = -3/2 and β = 3/2 are the zeroes of the polynomial.

As the given quadratic polynomial as we compared with ax² + bx + c

• a= 4

• b = 0

• c = -9

$\dag\:\underline{\underline{\bf{\blue{Sum\:of\:the\:zeroes\::}}}}$

$\longrightarrow\sf{\alpha +\beta =\dfrac{-b}{a} =\dfrac{Coefficient\:of\:x}{Coefficient\:of\:(x)^{2} } }\\\\\\\longrightarrow\sf{\dfrac{-3}{2} +\dfrac{3}{2} =\dfrac{-(0)}{4} }\\\\\\\longrightarrow\sf{\dfrac{-3+3}{2} =0}\\\\\\\longrightarrow\sf{\dfrac{0}{2} =0}\\\\\\\longrightarrow\sf{\pink{0=0}}$

$\dag\:\underline{\underline{\bf{\blue{Product\:of\:the\:zeroes\::}}}}$

$\longrightarrow\sf{\alpha \times \beta =\dfrac{c}{a} =\dfrac{Constant\:term}{Coefficient\:of\:(x)^{2} } }\\\\\\\longrightarrow\sf{\dfrac{-3}{2} \times\dfrac{3}{2} =\dfrac{-9}{4} }\\\\\\\longrightarrow\sf{\pink{\dfrac{-9}{4} =\dfrac{-9}{4} }}$

Thus;

Relationship between zeroes and coefficient is verified .