1. Unit 8 Seminar Agenda Solving Equations by Factoring Operations on Radical Expressions Complex Numbers

2. Quadratic Equation A quadratic equation has this format (standard form): ax 2 + bx + c = 0 Where a, b, and c are real numbers, and a cannot be 0 in order for an equation to be considered quadratic, there must be a squared term, and 2 has to be the highest power

3. Solving by factoring 1) Put the equation in standard form: ax 2 +bx+c = 0 2) Factor the quadratic 3) Set each factor equal to zero 4) Solve each equation 5) Check your answer.

4. Example: x 2 +x=12 Solve by Factoring. Put in standard form: x 2 +x-12=0 Factor: (x-3)(x+4) = 0 Set each factor equal to zero: x-3 = 0 or x+4 = 0 Solve: x = 3 or x = -4 Check: 3 2 +3-12 = 0 yes! (-4) 2 +(-4)-12 = 16-4-12 = 0 yes!

5. Try this one: 2x 2 -9x = 5

6. Try this one:

7. Simplifying Radicals Radicals must be in simplest form. Note: Learn what perfect squares are and perfect cubes – Perfect Squares: 1,4,9,16,25,36,49,64,81,100,121,144,169,196,225,…… Perfect Cubes: 1,8,27,64,125,216,343,512, 729,1000…… Example: Simplify:

8. Try this one:

9. Operations on Radical Expressions EXAMPLE1: To Add or Subtract Radicals: –1. Must have Same Index and Same Radicand –2. Add/Subtract coefficients

12. Try this one:

13. Rationalizing Denominators Simplifying Radicals: A radical is considered simplified when: The radical contains NO fractions and NO negative numbers NO radicals appear in the denominator of a fraction The technique we use to get rid of any radicals in the denominator of a fraction is called rationalizing.

14. To rationalizing denominators, we are going to multiply the denominator by something so that the index and the radicands exponent match meaning the radical(s) in denominator will simplify as an exponentless radicand. –Remember … if you multiply the denominator times something, you must multiply the numerator times the exact same thing!

15. EXAMPLE1:

16. Try this one:

17.  These examples had only ONE term in the denominator. If there are two terms, there is a slightly different technique required in order to rationalize the denominators.  We are going to multiply the denominator by its CONJUGATE (Remember … if you multiply the denominator times something, you must multiply the numerator times the exact same thing!)  CONJUGATES The conjugate of Notice the only difference between an expression and its conjugate is the arithmetic in the middle!

18.  EXAMPLE5:

19. Complex Numbers – see additional powerpoint!