Question

Write the family of quadratic polynomials having $-\frac{1}{4}$ and 1 as its zeros.

Answer 1

SOLUTION :

Let α  and β are the zeroes of the quadratic polynomial

Given :  α = -¼  & β = 1

α +  β = -¼ + 1 = (-1 +4)/4 = ¾

α +  β = ¾ …………….(1)

αβ = -¼ × 1 = -1/4

αβ =  -¼  ……………..(2)

Then, the quadratic polynomial is :  

k[x² –(sum of the zeroes)x + (product of the zeroes)]

k[x² –(α + β)x + (α β)]

=k[ x² - (¾) x + (-1/4)]

[From eq 1 & 2]

= k[ x² - 3x/4 - 1/4]

[K is any non zero real number]

Hence, the family of the quadratic polynomial is f(x) = k[ x² - 3x/4 - 1/4]

HOPE THIS ANSWER WILL HELP YOU..

Answer 2

Answer:

4x² - 3x - 1 = 0

Step-by-step explanation:

Given Zeroes are -1/4, 1.

Let the zeroes of the polynomial be α and β.

Let the polynomial be ax² + bx + c.

(i) Sum of Zeroes:

α + β = -b/a

-1/4 + 1 = -b/1

b = 3/4.

(ii) Product of Zeroes:

αβ = c/a

-1/4 * 1 = c/1

c = -1/4

Hence, the required quadratic polynomial = ax² + bx + c = 0

⇒ x² - (Sum of roots)x + (Product of roots) = 0

⇒ x² - (3/4)x + (-1/4) = 0

⇒ 4x² - 3x - 1 = 0

Hope it helps!