Find exact values of a and b if
Problem 1 :
3 + i is a root of x^{2} + 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 - i^{2}
= 10
Then, the required equations is
x^{2} - (sum of the roots)x + product of the roots = 0
x^{2} - 6x + 10
So, the values of a and b is -6 and 10.
Problem 2 :
1 - √2 is a root of x^{2} + 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
x^{2} - (sum of the roots)x + product of the roots = 0
x^{2} - 2x - 1
So, the values of a and b is -2 and -1.
Problem 3 :
a + ai is a root of x^{2} + 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)
= a^{2} - a(ai) + a(ai) - a^{2}i^{2}
= 2a^{2}
Then, the required equations is
x^{2} - (sum of the roots)x + product of the roots = 0
x^{2} + 4x + b = 0
2a = -4 ==> a = -2
2a^{2 }= 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 x^{2} + 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)
= √2^{2} - i^{2}
= 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 x^{2} - 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
a^{2} - (ai)^{2} = b
a^{2} - (a^{2}i^{2}) = b
a^{2} - a^{2}(-1) = b
a^{2} + a^{2 }= b
2a^{2} = b
2(-3)^{2} = b
b = 18
Feb 25, 24 07:44 AM
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