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Manipulate real and complex numbers and solve equations. AS 91577. Worksheet 1. Quadratics. General formula:. General solution:. Example 1. Equation cannot be factorised. Using quadratic formula. We use the substitution. A complex number. The equation has 2 complex solutions. Imaginary.
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Manipulate real and complex numbers and solve equations AS 91577
Quadratics General formula: General solution:
Example 1 Equation cannot be factorised.
Using quadratic formula We use the substitution A complex number
The equation has 2 complex solutions Imaginary Real
Adding complex numbers Subtracting complex numbers
(x + yi)(u + vi) = (xu – yv) + (xv + yu)i. Multiplying Complex Numbers
Conjugate If The conjugate of z is If The conjugate of z is
Solving by matching terms Match real and imaginary Real Imaginary
Solving polynomials Quadratics: 2 solutions 2 real roots 2 complex roots
If coefficients are all real, imaginary roots are in conjugate pairs
If coefficients are all real, imaginary roots are in conjugate pairs
Cubic Cubics: 3 solutions 3 real roots 1 real and 2 complex roots
Quartic Quartic: 4 solutions 2 real and 2 imaginary roots 4 real roots 4 imaginary roots
Solving a cubic This cubic must have at least 1 real solutions Form the quadratic. Solve the quadratic for the other solutions x = 1, -1 - i, 1 + i
Finding other solutions when you are given one solution. Because coefficients are real, roots come in conjugate pairs so Form the quadratic i.e. Form the cubic:
Just mark the spot with a cross
z = i z = -1 z =1 z = -i
Multiplying a complex number by a real number.(x + yi) u = xu + yu i.
Multiplying a complex number by i.zi = (x + yi) i = –y + xi.
Reciprocal of z Conjugate
Rectangular to polar form Using Pythagoras Modulus is the length Argument is the angle Check the quadrant of the complex number
Example 1 Rectangular form Polar form
Multiplying numbers in polar form Example 1
Multiplying numbers in polar form Example 2 Take out multiples of