Question:
Which quadratic equation has the roots -1+4i and -1-4i
A. X^2+2x+2=0
B. 2x^2+x+17=0
C. X^2+2x+17=0
D. 2x^2+x+2=0
Answer:
Option C
The quadratic equation that has roots -1 + 4i and -1 - 4i is [tex]x^{2}+2 x+17=0[/tex]
Solution:
Given, roots of a quadratic equation are (- 1 + 4i) and (- 1 – 4i)
We have to find the quadratic equation with above roots.
Now, as (-1 + 4i) and (-1 – 4i) are roots, x – (-1 + 4i) and x – (-1 – 4i) are factors of quadratic equation.
Then, equation will be product of its factors.
[tex](x-(-1+4 i)) \times(x-(-1-4 i))=0[/tex]
On multiplying each term with the terms in brackets we get,
[tex]x^{2}+x+4 i x+x+1+4 i-4 i x-4 i-16 i^{2}[/tex]
4ix and -4ix will cancel out each other.
Similarly 4i and -4i will cancel out each other
We know that [tex]i^2 = -1[/tex]
Hence we get,
[tex]x^{2}+2 x+1+16=0[/tex]
[tex]x^{2}+2 x+17=0[/tex]
Thus [tex]x^{2}+2 x+17=0[/tex] is the required quadratic equation
