Complex Numbers in Quadratic Equations

Solving quadratics The real numbers and imaginary numbers are subsets of the complex numbers So now we can say that all quadratic equations have two roots (possibly repeated) The roots will be complex (non-real) if the discriminant, b - 4ac < 0

Examples Determine the nature of the roots of these equations a. x + 4x 5 = 0 b. x 4x + 5 = 0 c. x + 2x + 1 = 0

Solve these quadratic equations with complex roots: i. ii. iii. x 4x + 13 = 0 x 2x + 6 = 0 x - 2x + 33 = 0 What do you notice about the pairs of solutions? Why does this happen? Irrational Roots

The complex conjugate For any complex number z = a + bi, there is a complex conjugate z* = a bi If z = 1 + 3i and w = -6 7i, find: a) z* b) w* c) (z w)*

Properties of complex conjugates: Properties of complex conjugates: For any complex number z = a + bi, a, b real What are the results for the following? z + z* z z* z.z*

Properties of complex conjugates: Properties of complex conjugates: For any complex number z = a + bi, a, b real z + z* = 2a purely real z z* = 2bi purely imaginary z.z* = a + b purely real We have seen the usefulness of the last result in division of complex numbers

Complex roots of quadratics The quadratic equation z2 2z + 5 = 0 What are the solutions to this? z = 1 2i The roots of this equation are 1 + 2i and 1 - 2i - a conjugate pair. This gives a sum of 2 and product 5, both real.

Conjugate pairs of roots It follows that: If a polynomial equation has real coefficients, then the complex roots occur in conjugate pairs.

One root of the quadratic equation ?2+ ?? + ? = 0, where p and q are real numbers, is 3 + 4i a) Write down the other root of the equation b) Find the value of p and the value of q FP1 textbook: Ex2.8 pp48-49 Q1-6 Q4-6 solutions