Question

Alapha and beta are zeroes if 4x2+4x+1 then form quadratic equation whose zeroes are 2alpha and 2beta

Answer 1

Answer:-

Given Polynomial : 4x² + 4x + 1

Let a = 4 , b = 4 , c = 1.

We know that,

sum of the zeroes = - b/a

→ α + β = - 4/4

→ α + β = - 1 -- equation (1)

Sum of the zeroes = c/a

→ αβ = 1/4 -- equation (2)

We have to find:

The equation whose zeroes are 2α and 2β.

General form of a quadratic equation is x² - (sum of the zeroes)x + product of the zeroes = 0

→ x² - (2α + 2β)x + (2α)(2β) = 0

→ x² - 2(α + β)x + 4(αβ) = 0

Putting the values from equation (1) & (2) we get,

→ x² - 2( - 1)x + 4 * (1/4) = 0

→ x² + 2x + 1 = 0

Hence, the required quadratic equation is x² + 2x + 1 = 0.

Answer 2

Step-by-step explanation:

Sum of zeros = -b/a

alpha + beta = -4/4

alpha + beta = -1

alpha = -1/beta

Product of zeros = c/a

alpha(beta) = 1/4

Polynomial:

k [x² - (sum of zeros)x + Product of zeros]

Where, k = 1

x² - (-1)x + 1/4

As said in question, we have to find the quadratic equation whose zeroes are 2alpha and 2beta.

→ x² - 2(-1)x + 4(1/4)

→ x² + 2x + 1

Hence, the polynomial is x² + 2x + 1.