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04Class 11 Maths

Complex Numbers and Quadratic Equations formulas

Complex number and quadratic formulas. Most problems get easier when you write z as a + ib, or in polar form.

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All Maths formulas

Complex numbers

Powers of i
i2=−1,i3=−i,i4=1\displaystyle i^2 = -1, \quad i^3 = -i, \quad i^4 = 1
Modulus
∣z∣=a2+b2\displaystyle |z| = \sqrt{a^2 + b^2}

For z = a + ib.

Conjugate
zˉ=a−ib,zzˉ=∣z∣2\displaystyle \bar z = a - ib, \quad z\bar z = |z|^2
Inverse
z−1=zˉ∣z∣2\displaystyle z^{-1} = \frac{\bar z}{|z|^2}
Polar form
z=r(cos⁡θ+isin⁡θ)\displaystyle z = r(\cos\theta + i\sin\theta)
Modulus rules
∣z1z2∣=∣z1∣∣z2∣,∣z1+z2∣≤∣z1∣+∣z2∣\displaystyle |z_1z_2| = |z_1||z_2|, \quad |z_1 + z_2| \le |z_1| + |z_2|

Quadratic equations

Roots
x=−b±b2−4ac2a\displaystyle x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
Sum and product of roots
α+β=−ba,αβ=ca\displaystyle \alpha + \beta = -\frac{b}{a}, \quad \alpha\beta = \frac{c}{a}
Discriminant
D=b2−4ac\displaystyle D = b^2 - 4ac

D < 0 gives complex conjugate roots.

Where students slip up

  • Writing √(−4) × √(−9) = √36. For negatives, use √(−a) = i√a first, which gives −6.
  • Taking the principal argument without checking the quadrant.

Try it on your weakest chapter.

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