Solving Equations with Complex Roots (Edexcel A-Level Further Mathematics): Revision Notes

📚 Revision Notes
Revision NotesModel AnswersQuizzesFlashcardsExam Questions

1.1.2 Solving Equations with Complex Roots

Overview

This topic explores solving quadratic and higher-order equations that yield complex roots, typically when the discriminant is negative. Understanding this is key to tackling equations involving imaginary components.

Solving Quadratic Equations with Complex Roots

A quadratic equation takes the form:

ax2+bx+c=0ax^2 + bx + c = 0

Using the quadratic formula:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Discriminant (Δ\Delta):

Δ=b2−4ac\Delta = b^2 - 4ac
  • If Δ>0\Delta > 0: Roots are real and distinct.
  • If Δ=0\Delta = 0: Roots are real and equal.
  • If Δ<0\Delta < 0: Roots are complex conjugates.
infoNote

Example 1: Solve x2+4x+5=0x^2 + 4x + 5 = 0

Step 1: Calculate the Discriminant

Δ=42−4(1)(5)\Delta = 4^2 - 4(1)(5) =16−20=:highlight[−4]= 16 - 20 = :highlight[-4]

Step 2: Apply the Quadratic Formula

x=−4±−42(1)x = \frac{-4 \pm \sqrt{-4}}{2(1)} =−4±2i2= \frac{-4 \pm 2i}{2}

Step 3: Simplify

x=−2±ix = -2 \pm i

Solution: x=:highlight[−2+i]x = :highlight[-2 + i] and x=:highlight[−2−i]x = :highlight[-2 - i]

infoNote

Complex roots always come in conjugate pairs.

Solving Higher-Order Polynomials

For equations of degree 3 or higher, complex roots may also arise. These can be solved using techniques such as factoring, synthetic division, or numerical methods.

infoNote

Example 2: Solve x2−6x+10=0x^2 - 6x + 10 = 0

Step 1: Calculate the Discriminant

Δ=(−6)2−4(1)(10)\Delta = (-6)^2 - 4(1)(10) =36−40=:highlight[−4]= 36 - 40 = :highlight[-4]

Step 2: Apply the Quadratic Formula

x=−(−6)±−42(1)x = \frac{-(-6) \pm \sqrt{-4}}{2(1)} =6±2i2= \frac{6 \pm 2i}{2}

Step 3: Simplify

x=3±ix = 3 \pm i

Solution: x=:highlight[3+i]x = :highlight[3 + i] and x=:highlight[3−i]x = :highlight[3 - i]

Note Summary

infoNote

Common Mistakes:

  1. Incorrect Discriminant Calculation: Forgetting that Δ=b2−4ac\Delta = b^2 - 4ac, leading to errors in identifying root types.
  2. Misapplying the Quadratic Formula: Mixing signs in −b±Δ-b \pm \sqrt{\Delta}
  3. Ignoring Conjugate Pairs: Failing to recognise that complex roots should occur as a+bia + bi and a−bia - bi
  4. Incomplete Simplification: Leaving roots in non-simplified forms.
  5. Misinterpreting Complex Roots: Confusing i2=−1i^2 = -1
infoNote

Key Formulas:

  1. Quadratic Formula:
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
  1. Discriminant:
Δ=b2−4ac\Delta = b^2 - 4ac
  1. Roots for Δ<0\Delta < 0:
x=a±bi, where i=−1x = a \pm bi, \text{ where } i = \sqrt{-1}
  1. Complex Conjugates: If z=a+biz = a + bi, then z‾=a−bi\overline{z} = a - bi

Explore Edexcel A-Level Further Mathematics Model Answers by Topics

Complex Numbers & Argand Diagrams

Exponential Form & de Moivre's Theorem

Explore Edexcel A-Level Further Mathematics Quizzes by Topics

Complex Numbers & Argand Diagrams

Exponential Form & de Moivre's Theorem

Explore Edexcel A-Level Further Mathematics Flashcards by Topics

Complex Numbers & Argand Diagrams

Exponential Form & de Moivre's Theorem

Explore Edexcel A-Level Further Mathematics Exam Questions by Topics

Complex Numbers & Argand Diagrams

Exponential Form & de Moivre's Theorem

Join 100,000+ A-Level students studying Revision Notes with us.

Select your subjects, and get access to A+ resources today.