Question

Obtain zeroes of 4√3x² + 5x - 2√3 and verify relation between its zeroes and coefficients.

Answer 1

Firstly factorise the given polynomial and then put each factor equal to zero to find required zeroes and then for verification show that

Sum of zeroes = - coefficient of x/coefficient of x²
Product of zeroes = constant term/coefficient of x²

SOLUTION:
Let p(x) = 4√3x² +5x -2√3
4√3x² +8x -3x -2√3

[By splitting the middle term]
4x (√3x +2) - √3(√3x +2)
(4x-√3) (√3x +2)

To find zeros, put p(x)= 0
(4x-√3)= 0 or  (√3x +2)= 0
4x = √3    or √3x = -2
x= √3/4   or x = -2/√3

Hence, zeroes of the polynomial are √3/4 and -2/√3.

Verification:
Sum of zeroes = (√3/4) +(-2/√3)
- coefficient of x/coefficient of x² =√3/4 -2/√3
- 5/4√3 = (√3×√3)-(2×4)/4√3
-5/4√3 =(3-8)/4√3
- 5/4√3 =-5/4√3

Product of zeroes = (√3/4) (-2/√3)= -½
constant term/coefficient of x² = -½
-2√3/ 4√3 = -½
-½ = -½

So, the relationship between the zeroes and its coefficients is verified.

HOPE THIS WILL HELP YOU...

Answer 2

Heya buddy !
Here's your answer :-

4√ 3 x^2 + 5x -2√3 (splitting the middle term)

=4√3 x^2 +(8-3)x -2√3

=4√3(√3x+2)-√3(√3x+2)

= (√3x+2) (4x-3)

x = -2/√3
x = √3/4

So, alpha + beta = -b/a
= -5/4√3

Sum of zeros = -2/√3 + √3/4 
= -5/4√3

Alpha×beta = c/a

= -2√3/4√3 

= -1/2
Product of zeros = -2/√3 × √3/4

= -2√3/4√3 

= -1/2

Since the products and sum of zeros are same. Hence, verified.

Hope this'll help uh (: