Question

x^2-(2+i)x-(1-7i)=0​

Answer 1

Answer:

The two solutions are x = 3 - i and x = -1 + 2i.

Step-by-step explanation:

The quadratic formula tells us that the roots of ax²+bx+c=0 are given by

 ( -b ± √Δ ) / (2a)

where Δ is the discriminant Δ = b² - 4ac.

For the given equation

 x² - ( 2 + i ) x - ( 1 - 7i ) = 0,

we have a = 1, b = - ( 2 + i ) and c = - ( 1 - 7i ).

So the discriminant is

 Δ = ( 2 + i )² + 4 ( 1 - 7i )

    = 4 - 1 + 4i + 4 - 28i

    = 7 - 24i

We want the square root of this.  That is, we want u and v so that

  ( u + vi )² = 7 - 24i

⇒ u² - v² + 2uv = 7 - 24i

⇒ u² - v² = 7 and uv = -12

⇒ u = 4, v = -3   or   u = -4, v = 3.

So one of the square roots is 4 - 3i.

Using this in the quadratic formula, the roots of the original equation are

 ( - b ±√Δ ) / 2a

= ( 2 + i ± ( 4 - 3i ) ) / 2

The two roots are therefore

 ( 2 + i + 4 - 3i ) / 2 = ( 6 - 2i ) / 2 = 3 - i

and

 ( 2 + i - 4 + 3i ) / 2 = ( -2 + 4i ) / 2 = -1 + 2i.

Hope this helps!