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Algebra - Complex Numbers and Quadratic Equations

Grade 11ICSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Introduction to Complex Numbers: A complex number zz is expressed in the form a+iba + ib, where aa is the real part and bb is the imaginary part. The imaginary unit ii is defined as −1\sqrt{-1}, such that i2=−1i^2 = -1. This allows for the square roots of negative numbers to be handled mathematically.

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The Argand Plane: Visually, complex numbers are represented on a 2D coordinate system called the Argand Plane. The horizontal x-axis represents the Real part (Re), and the vertical y-axis represents the Imaginary part (Im). A complex number z=a+ibz = a + ib is plotted as a unique point (a,b)(a, b) or represented as a position vector originating from (0,0)(0,0) to (a,b)(a, b).

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Modulus and Conjugate: The modulus ∣z∣|z| represents the distance of the point (a,b)(a, b) from the origin in the Argand plane. The conjugate of z=a+ibz = a + ib, denoted as zˉ=a−ib\bar{z} = a - ib, is visually the reflection of the point zz across the real axis (x-axis).

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Polar Representation: A complex number can be expressed using its distance from the origin rr (modulus) and the angle θ\theta (argument or amplitude) it makes with the positive real axis. Visually, this corresponds to circular coordinates r(cos⁡θ+isin⁡θ)r(\cos \theta + i\sin \theta), where θ\theta is measured counter-clockwise.

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Quadratic Equations and the Discriminant: For a quadratic equation ax2+bx+c=0ax^2 + bx + c = 0, the nature of roots is determined by the discriminant D=b2−4acD = b^2 - 4ac. If D<0D < 0, the equation has no real roots; instead, it has two complex roots that are conjugates of each other.

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Algebraic Operations: Addition and subtraction of complex numbers are performed by combining real parts with real parts and imaginary parts with imaginary parts, similar to vector addition. Multiplication involves the distributive property (FOIL) and substituting i2=−1i^2 = -1, while division requires multiplying the numerator and denominator by the conjugate of the denominator to rationalize it.

📐Formulae

i=−1,i2=−1,i3=−i,i4=1i = \sqrt{-1}, i^2 = -1, i^3 = -i, i^4 = 1

Standard Form: z=a+ibz = a + ib

Modulus: ∣z∣=a2+b2|z| = \sqrt{a^2 + b^2}

Conjugate: zˉ=a−ib\bar{z} = a - ib

Argument: θ=tan⁡−1(ba)\theta = \tan^{-1}(\frac{b}{a})

Polar Form: z=r(cos⁡θ+isin⁡θ)z = r(\cos \theta + i\sin \theta), where r=∣z∣r = |z|

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

Roots when D<0D < 0: x=−b±i∣D∣2ax = \frac{-b \pm i\sqrt{|D|}}{2a}

💡Examples

Problem 1:

Find the modulus and the principal argument of the complex number z=1+i3z = 1 + i\sqrt{3}.

Solution:

  1. Identify aa and bb: Here a=1a = 1 and b=3b = \sqrt{3}.
  2. Calculate the modulus: ∣z∣=a2+b2=12+(3)2=1+3=4=2|z| = \sqrt{a^2 + b^2} = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2.
  3. Calculate the argument: tan⁡θ=ba=31=3\tan \theta = \frac{b}{a} = \frac{\sqrt{3}}{1} = \sqrt{3}.
  4. Since both aa and bb are positive, the point lies in the first quadrant. Thus, θ=π3\theta = \frac{\pi}{3} or 60∘60^{\circ}.

Explanation:

The modulus is the magnitude of the vector representing the complex number, and the argument is the angle it makes with the positive real axis.

Problem 2:

Solve the quadratic equation x2+3x+5=0x^2 + 3x + 5 = 0.

Solution:

  1. Identify coefficients: a=1,b=3,c=5a = 1, b = 3, c = 5.
  2. Find the discriminant: D=b2−4ac=32−4(1)(5)=9−20=−11D = b^2 - 4ac = 3^2 - 4(1)(5) = 9 - 20 = -11.
  3. Since D<0D < 0, roots are complex: x=−b±D2a=−3±−112(1)x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-3 \pm \sqrt{-11}}{2(1)}.
  4. Use i=−1i = \sqrt{-1}: x=−3±i112x = \frac{-3 \pm i\sqrt{11}}{2}.
  5. Final roots: x=−32+i112x = -\frac{3}{2} + i\frac{\sqrt{11}}{2} and x=−32−i112x = -\frac{3}{2} - i\frac{\sqrt{11}}{2}.

Explanation:

We use the quadratic formula. Since the discriminant is negative, we introduce the imaginary unit ii to express the square root of the negative number.