FINDING UNKNOWN THROUGH SUM AND PRODUCT OF ROOTS

Problem 1 :

Find k if the difference between the roots of the quadratic equation

x² – 4x + k = 0 is 2.

Solution :

x² – 4x + k = 0

a = 1, b = -4, c = k

α – β = 2 ----- (1)

Products of the roots = αβ = c/a

3 ⋅ β = k/1

β = k/3

α – β = 2

3 – (k/3) = 2

-k/3 = 2 – 3

-k/3 = -1

k = 3

So, the value of k is 3.

Problem 2 :

Find the value of p such that the difference of the roots of the equation

x² – px + 8 = 0 is 2.

Solution :

x² – px + 8 = 0

a = 1, b = -p, c = 8

Difference between the roots = α – β = 2 ----- (1)

Sum of the roots = α + β = -b/a

α + β = -(-p)/1

α + β = p ----- (2)

Product of roots = α β = 8

(1) + (2)

2α = p + 2

α = (p + 2)/2

Applying the value of α = (p + 2)/2 in α β = 8

β = 8 ⋅ 2/(p + 2) 

β = 16/(p + 2)

Applying the values of α and β in (1), we get

(p + 2)/2 - 16/(p + 2) = 2

So, the value of p is -6 or 6.

Problem 3 :

Find the value of k such that the difference of the roots of the equation

2kx² – 20x + 21 = 0 is 2.

Solution :

2kx² – 20x + 21 = 0

a = 2k, b = -20, c = 21

Difference between the roots = α – β = 2 ----- (1)

α = 2 + β

Applying the value of α in (2) and (3), we get

Problem 4 :

Find k so that one root of the equation 2x² – 16x + k = 0 is twice the other. (Hint : One root = α, Other root = 2α)

Solution :

Given, 2x² – 16x + k = 0

a = 2, b = -16, c = k

α = α, β = 2α

By applying the value of α, we get

(8/3)² = k/4

64/9 = k/4

k = (64 ⋅ 4)/9

k = 256/9

So, the value of k is 256/9.

Problem 5 :

Find k so that one root of the equation

k(x – 1)² = 5x – 7

is twice the other.

Solution :

k(x – 1)² = 5x – 7

k(x² + 1 – 2x) – 5x + 7 = 0

kx² + k – 2xk – 5x + 7= 0

x²(k) - x(2k + 5) + k + 7 = 0

a = k, b = 2k + 5, c = k + 7

α = α, β = 2α

By applying α in (2), we get

Equating each factor to zero, we get

k = -25 and k = 2

Problem 6 :

Find k so that the roots of the quadratic equation

2x² + 3x + k = 0

are equal. 

Solution :

Given, 2x² + 3x + k = 0

a = 2, b = 3, c= k

α = α, β = α

So, the value of k is 9/8.

Problem 7 :

If 1 – i and 1 + i are the roots of the equation

x² + ax + b = 0

where a, b ∈  r, then find the values of a and b.

Solution :

Given, x² + ax + b = 0

a = 1, b = a, c = b

α = 1 - i, β = 1 + i

So, the values of a and b is -2 and 2 respectively.