Solving Equations with Complex Roots (Edexcel A Level Further Maths): Revision Note

Solving Quadratic Equations with Complex Roots

What are complex roots? 

• Complex numbers provide solutions for quadratic equations which have no real roots

• Complex roots occur when solving a quadratic with a negative discriminant

  • This leads to square rooting a negative number

How do we solve a quadratic equation with complex roots?

• We solve an equation with complex roots in the same way we solve any other quadratic equations

  • If in the form we can rearrange to solve

  • If in the form we can complete the square or use the quadratic formula

• We use the property along with a manipulation of surds

  • 

• When the coefficients of the quadratic equation are real, complex roots occur in complex conjugate pairs

  • If is a root of a quadratic with real coefficients then  is also a root

• When the coefficients of the quadratic equation are non-real, the solutions will not be complex conjugates

  • To solve these use the quadratic formula

How do we find a quadratic equation given a complex root?

• We can find the equation of the form  if you are given a complex root in the form 

  • We know that the complex conjugate is another root, 

  • This means that and are factors of the quadratic equation

  • Therefore 

    • Writing this as will speed up expanding

  • Expanding and simplifying gives us a quadratic equation where and are real numbers

Solving Polynomial Equations with Complex Roots

How many roots should a polynomial have?

• We know that every quadratic equation has two roots (not necessarily distinct or real)

• This is a particular case of a more general rule:

  • Every polynomial equation, with real coefficients, of degree n has n roots

  • The n roots are not necessarily all distinct and therefore we need to count any repeated roots that may occur individually

• From the above rule we can state the following:

  • A cubic equation of the form  can have either:

    • 3 real roots

    • Or 1 real root and a complex conjugate pair

• A quartic equation of the form  will have one of the following cases for roots:

  • 4 real roots

  • 2 real and 2 nonreal (a complex conjugate pair)

  • 4 nonreal (two complex conjugate pairs)

• When a real polynomial of any degree has one complex root it will always also have the complex conjugate as a root

Number of roots of a cubic function

Number of roots of a quartic function

How do we solve a cubic equation with complex roots?

• Steps to solve a cubic equation with complex roots

  • If we are told that is a root, then we know  is also a root

  • This means that  and  are factors of the cubic equation

  • Multiply the above factors together gives us a quadratic factor of the form 

  • We need to find the third factor 

  • Multiply the factors and equate to our original equation to get

• From there either

  • Expand and compare coefficients to find 

  • Or use polynomial division to find the factor 

• Finally, write your three roots clearly

How do we solve a polynomial of any degree with complex roots?

• When asked to find the roots of any polynomial when we are given one, we use almost the same method as for a cubic equation

  • State the initial root and its conjugate and write their factors as a quadratic factor (as above) we will have two unknown roots to find, write these as factors (z - α) and (z - β)

  • The unknown factors also form a quadratic factor (z - α)(z - β)

  • Then continue with the steps from above, either comparing coefficients or using polynomial division

    • If using polynomial division, then solve the quadratic factor you get to find the roots α and β

How do we solve polynomial equations with unknown coefficients?

• Steps to find unknown variables in a given equation when given a root:

  • Substitute the given root p + qi into the equation f(z) = 0

  • Expand and group together the real and imaginary parts (these expressions will contain our unknown values)

  • Solve as simultaneous equations to find the unknowns

  • Substitute the values into the original equation

  • From here continue using the previously described methods for finding other roots for the polynomial

How do we factorise a polynomial when given a complex root?

• If we are given a root of a polynomial of any degree in the form z  = p + qi

  • We know that the complex conjugate, z* = p – qi is another root

  • We can write (z – (p + qi)) and ( z – (p - qi)) as two linear factors

    • Or rearrange into one quadratic factor

  • This can be multiplied out with another factor to find further factors of the polynomial

• For higher order polynomials more than one root may be given

  • If the further given root is complex then its complex conjugate will also be a root

  • This will allow you to find further factors