Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
System of polynomial equations wikipedia , lookup
Elementary algebra wikipedia , lookup
Bra–ket notation wikipedia , lookup
Invariant convex cone wikipedia , lookup
Oscillator representation wikipedia , lookup
Quartic function wikipedia , lookup
Root of unity wikipedia , lookup
Quadratic form wikipedia , lookup
Cubic function wikipedia , lookup
Factorization wikipedia , lookup
Quadratic equation wikipedia , lookup
Using the Quadratic Formula to Find Complex Roots (Including Complex Conjugates) Complex Number A number consisting of a real and imaginary part. Usually written in the following form (where a and b are real numbers): a bi Example: Solve 0 = 2x2 – 2x + 10 x 2 2 2 4110 21 a = 1 b = -2 c = 10 2 36 2 1 3i and 1 3i 2 6i 2 1 3i Classifying the Roots of a Quadratic Describe the amount of roots and what number set they belong to for each graph: 1 Repeated 22Real Complex Roots Real Root Roots because because it has because it has it has twono one x-intercept x-intercepts (bounces off) A Quadratic ALWAYS has two roots Determining whether the Roots are Real or Complex What part of the Quadratic Formula determines whether there will be real or complex solutions? Discriminant < 0 b b 4ac x 2a 2 Complex Conjugates a bi The Complex Conjugate is: a bi For any complex number: The sum and product of complex conjugates are always real numbers Example: Find the sum and product of 2 – 3i and its complex conjugate. Complex Conugate: 2 3i Sum: 2 3i 2 3i 4 2 Product: 2 3i 2 3i 4 6i 6i 9i 4 9 13 Complex Roots Are Complex Conjugates A quadratic equation y = ax2 + bx + c in which b2 – 4ac < 0 has two roots that are complex conjugates. Example: Find the zeros of y = 2x2 + 6x + 10 6 2 4 2 10 2 4 x 6 x x 6 44 4 6 2 11i 4 x 3 11i 2 x 3 2 11 2 i 11 2 i and x 3 2 Complex Conjugates!