Which quadratic equation has roots - 1 + 4i and - 1 - 4i?. (1 point)A. x^2+2x+2=0B. 2x^2+x+17=0C. x^2+x+2=0D. 2x^2+x+2=0

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=0Answer:Option CThe 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 otherWe 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