Question

If alpha and beta are the zeroes of the polynomial x^2-3x+k , such that alpha-beta=1, find the value of k

Answer 1

Question:

If α and ß are the zeros of the given polynomial

x² - 3x + k , such that α - ß = 1 , then find the value of k .

Answer:

k = 2

Note:

• A polynomial of degree 2 is called quadratic polynomial.

• The general form of a quadratic polynomial is given by ; ax² + bx + c = 0 .

• The possible values of variable ( or unknown ) for which the polynomial becomes zero are known as its zeros.

• If α and ß are the zeros of the quadratic polynomial ax² + bx + c , then ;

Sum of zeros , (α + ß) = -b/a

Product of zeros , α•ß = c/a

• If α and ß are the zeros of any quadratic polynomial then that polynomial is given as ;

x² - (α + ß)x + α•ß .

Solution:

Here ,

The given quadratic polynomial is ; x² - 3x + k

Clearly , we have ;

a = 1

b = -3

c = k

Now,

The sum of the zeros of the given quadratic polynomial is ;

=> α + ß = -b/a

=> α + ß = -(-3)/1

=> α + ß = 3 ------(1)

Also,

The product of zeros of the given quadratic polynomial is ;

=> α•ß = c/a

=> α•ß = k/1

=> α•ß = k -------(2)

Also,

It is given that ,

α - ß = 1 ------(3)

Now,

We know that ; (x + y)² - (x - y)² = 4xy

Thus,

=> (α + ß)² - (α - ß)² = 4αß

=> 3² - 1² = 4k

=> 4k = 3² - 1²

=> 4k = 9 - 1

=> 4k = 8

=> k = 8/4

=> k = 2

Hence,

The required value of k is 2 .

Answer 2

Answer:-

Given polynomial => x² - 3x + k = 0

$\alpha - \beta = 1 \: - \: equation \: (1)$

We know that,

$sum \: of \: the \: roots \: ( \alpha + \beta ) = \frac{ - b}{a} \\ \\ \alpha + \beta = \frac{ - ( - 3)}{1} \\ \\ \alpha + \beta = 3 \: - \: equation \: (2)$

Add both the equations,

$2 \alpha = 3 + 1 \\ \\ 2 \alpha = 4 \\ \\ \alpha = \frac{4}{2} = 2$

$substitute \: \alpha \: value \: in \: equation \: (1) \\ \\ \alpha - \beta = 1 \\ \\ 2 - \beta = 1 \\ \\ \beta = 1$

Substitute , the roots in the given polynomial.

(2)² - 3(2) + k = 0

4 - 6 + k = 0

k = 2

(Or)

(1)² - 3(1) + k = 0

1 - 3 + k = 0

k = 2

Hence, the value of k is 2.