Question

Find the quadratic polynomial, sum of whose zeros is 8 and their product is 12. Also find the zeroes of the polynomial.

Answer 1

GIVEN :

Let the zeroes be α and β

Sum of zeroes of a quadratic equation(α+β)

= 8

Product of zeroes (αβ) = 12

Quadratic equation =?

We know that,

Quadratic equation = x² - (sum of zeroes)x + product of zeroes

=x² - (α+β) + αβ

=> x² - (8)x + 12

=> x² - 8x + 12

Therefore, the quadratic equation is x²- 8x + 12

ZEROES :

x² - 8x + 12

Split the middle term

x² - 2x - 6x + 12

x(x - 2) - 6(x - 2)

x - 2 = 0 ; x - 6 = 0

x = 2 and x = 6

Therefore, the zeroes are 6 and 2.

Answer 2

Answer:

Let α and β be the roots of the quadratic equation.

ATQ,

α + β = 8..(1)

α β = 12..(2)

Let p(x) be the quadratic equation.

p(x) = x² + (α + β)x + α β

=> x² + 8x + 12

Thus, the quadratic polynomial is x² + 8x + 12.

Now, we know that,

(α - β)² = (α + β)² - 4 αβ

(α - β)² = (8)² - 4 × 12

(α - β)² = 64 - 48

(α - β)² = 16

(α - β) = 4..(3)

From (1) and (3),

α + β = 8 and α - β = 4

Subtracting (1) and (3) We get:

α = 6 and β = 2

Hence, Final answer: Zeroes of the polynomial are 6,2

Quadratic polynomial : x² + 8x + 12