Question

find the zeros of the quadratic polynomial 4x2 - 4x +1 and verify the relationship between the zeros and coeffcients

Answer 1

Hey Mate !

Here is your solution :

Given,

Quadratic eq. = 4x² - 4x + 1

Here,

Coefficient of x² ( a ) = 4

Coefficient of x ( b ) = -4

Constant term ( c ) = 1

Now,

=> 4x² - 4x + 1 = 0

=> ( 2x )² - 2 × 2x × 1 + ( 1 )² = 0

Using identity :

=> ( a² - 2ab + b² ) = ( a - b )²

=> ( 2x - 1 )² = 0

=> ( 2x - 1 ) ( 2x - 1 ) = 0 ( continued further )

=> ( 2x - 1 ) = 0 ÷ ( 2x - 1 )

=> ( 2x - 1 ) = 0

=> x = 1/2

★

=> ( 2x - 1 ) = 0 ÷ ( 2x - 1 )

=> ( 2x - 1 ) = 0

=> 2x = 1

=> x = 1/2

Hence, zeroes are ( 1/2 ) and ( 1/2 ).

Now,

=> Sum of zeroes = -b/a

=> ( 1/2 ) + ( 1/2 ) = -( -4 ) ÷ 4

=> ( 1 + 1 )/2 = 4 ÷ 4

=> 2÷2 = 1

=> 1 = 1

And,

=> Product of zeroes = c/a

=> ( 1/2 ) × ( 1/2 ) = 1/4

=> 1/4 = 1/4

★ Verified ★


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Hope it helps !! ^_^

Answer 2

Answer:

The zeros of the quadratic polynomial 4x² - 4x +1 are $\frac{1}{2}$,$\frac{1}{2}$

Step-by-step explanation:

Given,

Quadratic equation 4x² - 4x +1

To find,

1. The zeros of the quadratic polynomial
2. Verify the relationship between the zeros and coefficients.

Recall the concepts:

If 'α' and 'β' are the roots of the quadratic equation ax² + bx + c = 0,

Then the relation between the zeros and coefficients

sum of roots = α +β = $\frac{-b}{a}$

and Product of roots = αβ = $\frac{c}{a}$

(a-b)² = a² - 2ab +b²  -----------(A)

Solution:

4x² - 4x +1 = (2x)² - 2(2x)(1)+1², this is of the form a² - 2ab +b²

∴(2x)² - 2(2x)(1)+1² = (2x -1)²(from the identity (A))

To find the zeros

4x² - 4x +1  = 0 ⇒  (2x -1)² = 0 ⇒ x = $\frac{1}{2}$,$\frac{1}{2}$,

∴The zeros of the quadratic polynomial 4x² - 4x +1 are $\frac{1}{2}$,$\frac{1}{2}$

To verify the relationship between the zeros and coefficients.

Sum of zeros=  $\frac{1}{2}$ + $\frac{1}{2}$ = 1

Product of zeros = $\frac{1}{2}$ ×$\frac{1}{2}$ = $\frac{1}{4}$

Comparing the equation  4x² - 4x +1 with  ax² + bx + c = 0 we get,

a = 4, b = -4 and c= 1

$\frac{-b}{a}$  = $\frac{-(-4)}{4}$ = 1

$\frac{c}{a}$ = $\frac{1}{4}$

Sum of zeros =  $\frac{-b}{a}$ = 1

Product of zeros =  $\frac{c}{a}$ =$\frac{1}{4}$

∴ The relationship between the zeros and coefficients is verified.

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