Explore the world of complex numbers and master solving quadratic equations with real and imaginary solutions.
Complex Numbers and Quadratic Equations – Formula & Equation Sheet
Essential formulas and equations from Mathematics, tailored for Class 11 in Mathematics.
This one-pager compiles key formulas and equations from the Complex Numbers and Quadratic Equations chapter of Mathematics. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Key concepts & formulas
Essential formulas, key terms, and important concepts for quick reference and revision.
Formulas
z = a + ib
z is a complex number where a and b are real numbers. a represents the real part (Re z) and b is the imaginary part (Im z).
D = b² - 4ac
D is the discriminant of the quadratic equation ax² + bx + c = 0. It determines the nature of the roots: D > 0 (two distinct real roots), D = 0 (one real root), D < 0 (two complex roots).
z1 + z2 = (a + c) + i(b + d)
This formula shows the addition of two complex numbers z1 = a + ib and z2 = c + id.
z1 - z2 = (a - c) + i(b - d)
This denotes the difference of two complex numbers z1 = a + ib and z2 = c + id.
z1 * z2 = (ac - bd) + i(ad + bc)
This formula expresses the multiplication of two complex numbers z1 = a + ib and z2 = c + id.
z1 / z2 = (z1 * z2̅) / (z2 * z2̅)
To divide by a complex number, multiply by its conjugate. Here, z2̅ is the conjugate of z2.
|z| = √(a² + b²)
The modulus of a complex number z = a + ib is the distance from the origin in the Argand plane.
z̅ = a - ib
The conjugate of the complex number z = a + ib is denoted as z̅ and involves changing the sign of the imaginary part.
i² = -1
This defines the unit imaginary number, where i represents the square root of -1.
i⁴ = 1
This shows the periodicity of the powers of i with a cycle of 4.
Equations
ax² + bx + c = 0
The standard form of a quadratic equation, where a, b, and c are constants.
x = (-b ± √D) / 2a
The quadratic formula to find the roots of the equation ax² + bx + c = 0, where D is the discriminant.
z1 + z2 = z2 + z1
The commutative property of addition for complex numbers.
z1(z2 + z3) = z1z2 + z1z3
The distributive property of multiplication for complex numbers.
(z1 + z2)² = z1² + 2z1z2 + z2²
This is the expansion of the square of a sum of two complex numbers.
Re(z) = (z + z̅) / 2
This represents the real part of a complex number using its conjugate.
Im(z) = (z - z̅) / (2i)
This represents the imaginary part of a complex number using its conjugate.
z₁ z̅₁ = |z₁|²
The product of a complex number and its conjugate gives the square of its modulus.
z = r(cos θ + i sin θ)
The polar form of a complex number, where r is the modulus and θ is the argument.
D = b² - 4ac = 0
A specific case in the quadratic formula indicating that the equation has exactly one real root.
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