Question

2. Find the zeroes of each of the following quadratic
polynomials and verify the relationship between
the zeroes and their
coefficients.
(i)x^2+3/2root5-5​

Answer 1

Step-by-step explanation:

Given:-

x^2+3/2root5-5

To find:-

Find the zeroes of the following quadratic

polynomial and verify the relationship between

the zeroes and their coefficients ?

Solution:-

Given quadratic polynomial = X^2 +3/2 √5X -5

=> X^2 +3√5X/2 - 5

To get the zeores we write it

X^2 +3√5X/2 - 5 = 0

=>( 2X^2 +3√5X -10)/2 = 0

=> 2X^2+3√5X -10 = 0×2

=> 2X^2+3√5X -10 = 0

=> 2X^2+4√5X -√5X -10 = 0

=> 2X(X +2√5) -√5 (X +2√5) = 0

=> (X+2√5)(2X-√5) = 0

=> X+2√5 = 0 or 2X -√5 = 0

=> X = -2√5 or X = √5/2

Zeroes are -2√5 and √5/2

Relationship between zeroes and the coefficients:-

Given Polynomial = X^2 +3√5X/2 - 5

On Comparing this with the standard quadratic Polynomial ax^2+bx+c

a = 1

b = 3√5/2

c = -5

And the zeroes are -2√5 and √5/2

Let α = -2√5 and β = √5/2

i) Sum of the zeroes = -2√5 + √5/2

=>[ 2(-2√5)+√5]/2

=> (-4√5+√5)/2

=> -3√5/2

α + β = -b/a

=> -(3√5/2)/1

=> -3√5/2

Sum of the zeroes = -b/a

ii) Product of the zeroes

= (-2√5)(√5/2)

=-5

αβ = c/a

=> -5/1

=>-5

Product of the zeroes = c/a

Verified the given relations

Used formulae:-

• the standard quadratic Polynomial ax^2+bx+c
• Sum of the zeroes = α + β = -b/a
• Product of the zeroes = αβ = c/a