Question

If α and β are the zeroes of the polynomial x2+7x+3, then the value of (α-β)2

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Answer 1

Answer:

${( \alpha - \beta )}^{2} = {( \alpha + \beta )}^{2} - 4 \alpha \beta \\ = {( - 7)}^{2} - 4 \times 3 \\ = 49 - 12 \\ = 37$

Answer 2

Answer :

(α - ß)² = 37

Note:

★ The possible values of the variable for which the polynomial becomes zero are called its zeros .

★ A quadratic polynomial can have atmost two zeros .

★ The general form of a quadratic polynomial is given as ; ax² + bx + c .

★ 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 , (αß) = c/a

Solution :

Here ,

The given quadratic polynomial is ;

x² + 7x + 3 .

Now ,

Comparing the given quadratic polynomial with the general quadratic polynomial ax² + bx + c , we have ;

a = 1

b = 7

c = 3

Also ,

It is given that , α and ß are the zeros of the given quadratic polynomial .

Thus ,

The sum of zeros will be given as ;

=> α + ß = -b/a

=> α + ß = -7/1

=> α + ß = -7

Also ,

The product of zeros will be given as ;

=> αß = c/a

=> αß = 3/1

=> αß = 3

Also ,

We know that ,

(A - B)² = (A + B)² - 4AB

Thus ,

=> (α - ß)² = (α + ß)² - 4αß

=> (α - ß)² = (-7)² - 4•3

=> (α - ß)² = 49 - 12

=> (α - ß)² = 37

Hence ,

(α - ß)² = 37