Quadratic Equation Question?

Question

If a + bi is a root of a quadratic equation with real coefficients then a - bi is also a root.
This gives us a equation:
(x - (a + bi))(x - (a - bi)) = 0
=> x² - 2ax + (a² + b²) = 0.................(1)

Now given equation (1) we can easily find the roots of all equations.
METHOD A]
Example:
x² - 8x + 17 = 0
gives a = (-8) / (-2) = 4
and a² + b² = 17
=> 16 + b² = 17
=> b² = 1
=> b = ±1
Hence 4 + i and 4 - i are two roots of the equation

We all use formulas like x = (-b ± √D)/2a for finding roots (Applicable to ax² + bx + c = 0)
I want to verify whether the above method (mentioned in (1) and method (A)) has any limitations.

I want to make sure as I may use it for solving problems
Thanks for any answers and explanations

Anonymous · Accepted Answer

If a + bi is a root of a quadratic equation, then a - bi is also a root.

That's correct, because bi comes from the square root in the formula.
A square root has always ± solutions.

husoski · Answer

Sure, use it.  The only limitations are the two obvious ones: all coefficients are real, and the leading coefficient (on xÂ²) is 1.  It works better on complex roots than real ones, in a sense.  Take:

xÂ² - 2x - 24 = 0

You get a=(-2)/(-2) = 1 and b=â(-24 - 1Â²) = â-25 = Â±5i.  Now, the roots are:

1 Â± (5i)i = 1 - 5 and 1 + 5, or -4 and +6

See? To get real roots by this method you take a (very small) detour through the complex plane.

Rameshwar · Answer

this is not any thing new  as we know that 
 if  x^2 + mx + n =0 is  quadratic  equation then 
  m = - sum of roots  &  n= product of roots
    if  ( a+bi ) & (a-bi ) are two roots then 
  m= -2a  & n = a^2 +b^2 
so equation can be  ---
          x^2 - 2ax + (a^2 +b^2) =0
 you can solve equation by this method-----
  let equation is  x^2 - 8x + 17 =0 
 let   a  &  b are roots then 
  sum of roots  a+ b=  8 ---------------(I)
   product of roots ab = 17
 so  a- b= sqrt[ (a+b)^2 - 4ab ]
      a- b= sqrt [ 64- 68]= sqrt(-4)
      or a- b= 2 i------------------------(ii)
  solving (i)  & (ii)  a=  4+i   &  b=  4-i

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Quadratic Equation Question?

3 Answers