Solve the following quadratic equations:
x2 – (5 – i)x + (18 + i) = 0
x2 – (5 – i)x + (18 + i) = 0
Given x2 – (5 – i)x + (18 + i) = 0
Recall that the roots of quadratic equation ax2 + bx + c = 0, where a ≠ 0, are given by
Here, a = 1, b = –(5 – i) and c = (18 + i)
By substituting i2 = –1 in the above equation, we get
By substituting –1 = i2 in the above equation, we get
We can write 48 + 14i = 49 – 1 + 14i
⇒ 48 + 14i = 49 + i2 + 14i [∵ i2 = –1]
⇒ 48 + 14i = 72 + i2 + 2(7)(i)
⇒ 48 + 14i = (7 + i)2 [∵ (a + b)2 = a2 + b2 + 2ab]
By using the result 48 + 14i = (7 + i)2, we get
[∵ i2 = –1]
∴ x = 2 + 3i or 3 – 4i
Thus, the roots of the given equation are 2 + 3i and 3 – 4i.