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:

ax^2 + bx + c = 0

Using the quadratic formula:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Discriminant ( $\Delta$ ):

\Delta = b^2 - 4ac

If $\Delta > 0$ : Roots are real and distinct.
If $\Delta = 0$ : Roots are real and equal.
If $\Delta < 0$ : Roots are complex conjugates.

infoNote

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

Step 1: Calculate the Discriminant

\Delta = 4^2 - 4(1)(5)

= 16 - 20 = :highlight[-4]

Step 2: Apply the Quadratic Formula

x = \frac{-4 \pm \sqrt{-4}}{2(1)}

= \frac{-4 \pm 2i}{2}

Step 3: Simplify

x = -2 \pm i

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

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.

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

Step 1: Calculate the Discriminant

\Delta = (-6)^2 - 4(1)(10)

= 36 - 40 = :highlight[-4]

Step 2: Apply the Quadratic Formula

x = \frac{-(-6) \pm \sqrt{-4}}{2(1)}

= \frac{6 \pm 2i}{2}

Step 3: Simplify

x = 3 \pm i

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

Note Summary

infoNote

Common Mistakes:

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

infoNote

Key Formulas:

Quadratic Formula:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Discriminant:

\Delta = b^2 - 4ac

Roots for $\Delta < 0$ :

x = a \pm bi, \text{ where } i = \sqrt{-1}

Complex Conjugates: If $z = a + bi$ , then $\overline{z} = a - bi$

Solving Equations with Complex Roots Simplified Revision Notes for A-Level Edexcel Further Maths Core Pure

1.1.2 Solving Equations with Complex Roots

Overview

Solving Quadratic Equations with Complex Roots

Discriminant ( $\Delta$ ):

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

Solving Higher-Order Polynomials

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

Note Summary

Common Mistakes:

Key Formulas:

500K+ Students Use These Powerful Tools to Master Solving Equations with Complex Roots For their A-Level Exams.

Other Revision Notes related to Solving Equations with Complex Roots you should explore

Solving Equations with Complex Roots Simplified Revision Notes for A-Level Edexcel Further Maths Core Pure

1.1.2 Solving Equations with Complex Roots

Overview

Solving Quadratic Equations with Complex Roots

Discriminant (Δ\DeltaΔ):

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

Solving Higher-Order Polynomials

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

Note Summary

Common Mistakes:

Key Formulas:

500K+ Students Use These Powerful Tools to Master Solving Equations with Complex Roots For their A-Level Exams.

Other Revision Notes related to Solving Equations with Complex Roots you should explore

Discriminant ( $\Delta$ ):

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

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