FIND THE MISSING VALUES IN QUADRATIC EQUATION WITH COMPLEX ROOTS

Find exact values of a and b if 

Problem 1 :

3 + i is a root of x² + ax + b = 0, where a and b are real

Solution :

Since, a complex number 3 + i is one of the roots, then its conjugate 3 - i will be another roots.

Sum of the roots = (3 + i) (3 - i)  

= 6

Product of the roots = (3 + i) (3 - i)

= 9 - 3i + 3i - i²

= 10

Then, the required equations is 

x² - (sum of the roots)x + product of the roots = 0

x² - 6x + 10

So, the values of a and b is -6 and 10.

Problem 2 :

1 - √2  is a root of x² + ax + b = 0, where a and b are rational

Solution :

Since, a complex number 1 - √2 is one of the roots, then its conjugate 1 + √2 will be another roots.

Sum of the roots = (1 - √2) (1 + √2)  

= 2

Product of the roots = (1 - √2) (1 + √2)

= 1 + √2 - √2 - 2

= -1

Then, the required equations is 

x² - (sum of the roots)x + product of the roots = 0

x² - 2x - 1

So, the values of a and b is -2 and -1.

Problem 3 :

a + ai is a root of x² + 4x + b = 0, where a and b are real.

Solution :

Since, a complex number a + ai is one of the roots, then its conjugate a - ai will be another roots.

Sum of the roots = (a + ai) (a - ai)  

= 2a

Product of the roots = (a + ai) (a - ai)

= a² - a(ai) + a(ai) - a²i²

= 2a²

Then, the required equations is 

x² - (sum of the roots)x + product of the roots = 0

x² + 4x + b = 0

2a = -4 ==> a = -2

2a² = b

Applying the value of a, we get

2(2)² = b

b = 8

Problem 4 :

Find the exact values of a and b if √2 + i is a root of x² + ax + b = 0, where a and b are real numbers.

Solution :

Roots of the given quadratic polynomial are √2 + i and √2 - i.

Sum of roots = √2 + i + √2 - i = 2√2

Product of roots = (√2 + i) (√2 + i)

= √2² - i²

= 2 - (-1)

= 3

Sum of the roots = -a = 2√2 ==> a = -2√2

Product of roots = b = 3

Problem 5 :

a+ai is a root of x² - 6x + b = 0, where a and b are real numbers. Find the values of a and b.

Solution :

Roots of the given quadratic polynomial are a + ai and a - ai.

Sum of roots :

a + ai + a - ai = -6

2a = - 6

a = -3

Product of roots :

(a + ai)(a - ai) = b

a² - (ai)² = b

a² - (a²i²) = b

a² - a²(-1) = b

a² + a² = b

2a² = b

2(-3)² = b

b = 18