Question

If one zero of the polynomial x2+kx+18 is double the other zero, then k=?


a. 9
b. ±9
c. 3
d. ±3


If one zero of 2x2-3x+k is the reciprocal to the other find the value of k.


a. 2
b. -23
c. -3
d. -32

Answer 1

∆ Main Concept ;

# Relationship between Zeros and coefficiants of a Quadratic Polynomial :

For a qudratic polynomial of the Form ax² + bx + c

Sum of Zeros = $\sf-\dfrac{b}{a}$

Product of Zeros = $\sf\dfrac{c}{a}$

∆ Part - 1

Let First Zero be = α

According to the Given condition,

Second zero should be = 2α

Now,

Product = α × 2α = 18

⇒ 2α² = 18

⇒α² = 9

⇒ α = ±3

Now,

Case (1) :-

For α = 3

• 1st zero = 3

• 2nd zero = 6

Sum = 3 + 6 = -k

⇒ k = -9

Case (2) :-

For α = - 3

• 1st zero = -3

• 2nd zero = -6

Sum = -3 -6 = -k

⇒k = 9

Combining Both We Get,

k = ±9

So,

Option b is correct.

____________________

∆ Part - 2

Let First Zero be = α

According to the Given condition,

Second zero Should be = 1/α

Now,

Product = α × 1/α = k/2

⇒ 1 = k/2

⇒k = 2

Hence,

Option a is correct

Answer 2

Given :-

1] If one zero of the polynomial x² + kx + 18 is double the other zero

2] If one zero of 2x² - 3x + k is the reciprocal to the other

To Find :-

1] Value of k

2] Value of k

Solution :-

1]

Let the first zero be α and 2α

2α × α = 18/1

2α² = 18

α² = 18/2

α² = 9

α = √9

α = ±3

Now

Either value of α is 3 or -3

When α is 3

α = 3

2α = 2(3) = 6

Sum = 6 + 3 = 9

When α = -3

α = -3

2α = 2(-3) = -6

Sum = -3 + (-6) = -3 - 6 = -9

Option B is correct

2]

Let the zeroes be α and 1/α

α × 1/α = k/2

α/α = k/2

1 = k/2

1 × 2 = k

2 = k

Option A is correct