Question

Find the zeroes of the quadratic
polynomial & 4x^2 – 4 x-3 and verify the
relation between zeroes and coefficients​

Answer 1

Step-by-step explanation:

4x^2-4x-3=0

4x^2-6x+2x-3=0

2x(2x-3)+1(2x-3)=0

(2x-3)(2x+1)=0

2x-3=0

x=3/2

2x+1=0

x=-1/2

therefore the zeros of the given quadratic equation are

3/2 and -1/2

Answer 2

Zeroes of the polynomial are $x_{1}=3/2$  ,$x_{2}=-1/2$

Step-by-step explanation:

Given,

Polynomial    $f\left ( x \right )=4x^{2}-4x-3$

                                 $4x^{2}-4x-3=0$

Middle term split into two parts

$4x^{2}-6x+2x-3=0$

$2x \left (2x-3 \right )+1\left ( 2x-3 \right )=0$

$\left ( 2x-3 \right )\left ( 2x+1 \right )=0$

$2x-3=0$

$2x=3$

$x=3/2$

$x_{1}=3/2$

OR

$2x+1=0$

$2x=-1$

$x=-1/2$

$x_{2}=-1/2$

Let a polynomial

$f\left ( x \right )=ax^{2}+bx+c=0$     This polynomial have two zeroes $\alpha ,\beta$

sum of roots      = $\alpha +\beta =-b/a$

product of roots=$\alpha \times \beta =c/a$

compare of two equations

1.   Sum of roots

$\alpha +\beta =-\left ( -4 \right )/4=1$

$x_{1} +x_{2} =-1/2+3/2=1$

2.  Product of roots

$\alpha \times \beta =\left ( -3\right )/4=-3/4$

$x_{1} \times x_{2}=-1/2\times 3/2=-3/4$

Zeroes of polynomial is

$x_{1}=3/2$

$x_{2}=-1/2$