Sunday, March 31, 2013

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.

1. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
(i) x2 – 2x – 8             (ii) 4s2– 4s + 1          (iii) 6x2 – 3 – 7x
(iv) 4u2 + 8u              (v) t2 – 15                   (vi) 3x2x – 4
          
To factorize the value we have to find two value which
sum is equal to, b =-2
 and product is , a*c  = 1*-8 = -8
2 and -4 are required values which
sum is 2-4     = - 2
product is 2*-4 = - 8
So we can write middle term -2 x  = 2 x – 4 x 
We get
           
x (x +2) -4(x+2) = 0
(x+2)(x-4)           = 0
First zero
X+2                 = 0
X                     = - 2
Second zero
x-4                   = 0
x                      = 4
sum of zero              -2 + 4 = 2
product of zero         2*-4     = -8

For a equation ax2+ bx  + c  = 0 , if  zero are α and β ,

Plug the values of a , b and c we get
Sum  of zeros                       -b/a     = -(-2/1) = 2
Product of zeros                    c/a      =-8/1      = -8 
hence we have verified that

(ii) 4s2 – 4s + 1
Factorize the equation
  
Compare the equation with as2  + bs  + c  = 0
We get
a = 4 ,b=-4 c=1
To factorize the value we have to find two value which
sum is equal to         b          =-4
product is                   a*c      = 4
-2 and -2 are required values which
sum is              2   – 2         =  - 4   
product is       - 2 * - 2            =    4
So we can write middle term         - 4 s   = - 2 s  – 2 s   
We get
2s (2s -1) -1(2s-1) = 0
(2s-1)(2s-1)         = 0
Solve for first zero
2s-1    =0
2s        = 1
s          = ½
Solve for second zero
2s-1    =0
2s        = 1
S         = ½
sum of zeros    1 /2 + 1/2   =2/2 =  1
product of zeros   ½* ½     = 1/4
Sum of zero = -b/a = -(-4/4) = 1
Product of zero= c/a =  1/4 = 1/4
hence we have verified that 
(iii) 6x2 – 3 – 7x
Factorize the equation
  
Compare the equation with ax2  + bx  + c  = 0
We get
a = 6 ,b=-7 c= -3
To factorize the value we have to find two value which
sum is equal to         b =-7
product is       a*c  = 6*-3      = -18
2 and -9 are required values which
sum is             2 – 9     = - 7
product is 2 * - 9= - 18
So we can write middle term as
 -7x      = 2x – 9x 
We get
 
2x (3x +1) -3(3x +1) = 0
(3x+1)(2x-3)   = 0
3x+1 = 0 , 2x-3 = 0
Solve for first zero
3x = - 1
x  = -1/3
Solve for second zero
2x-3 = 0
2x    = 3
X      =  3/2
   
Plug the values of a , b and c we get
Sum  of zeros     -b/a     = -(-7/6)      = 7/6
Product of zero    c/a     =  -3/6         = -1/2
hence we have verified that  

(iv) 4u2 + 8u
Factorize the equation
Compare the equation with au2  + bu + c  = 0
We get
a = 4 ,b=-8 c= 0
To factorize the value  we can take 4u common there
4u(u+2)= 0
First zero
4u        = 0
        = 0  
second zero
u+2      = 0
u          = - 2  
sum of zero       0 - 2        = - 2
product of zero  0 * ( - 2 ) = 0

Sum  of zero , -b/a   = -(8/4)    = -2
Product of zero c/a =  0/4      =  0
hence we have verified that  
 
(vi) 3x2x – 4
 Factorize the equation
  
Compare the equation with
ax2  + bx  + c  = 0
We get
a = 3 ,b=-1 c= -4
to factorize the value we have to find two value which
sum is equal to  b    =-1
product is           a*c  = 3*-4  = -12
3 and -4 are required values which
Sum is           3 – 4   = - 1
Product is      3 *- 4   = - 12
So we can write middle term  -x  = 3x – 4x 
We get
3x (x +1) -4(x +1) = 0
(x + 1)( 3x - 4) = 0
x+1 = 0 , 3x-4   = 0
Solve for first zero
x  = - 1
solve for second zero
3x-4 = 0
3x = 4
x = 4/3
  

Sum of zeros  -b/a   = -(-1/3)= 1/3
Product of zero c/a   =  -4/3  = 4/3
hence we have verified that

 

4 comments: