2. Quadratic Equations You start with a quadratic equation: ax2+bx+c=0 where a, b, and c are real numbers ex. x2+5x+6=0
3. Factoring First step is to try to factor. We want tofind: 2 numbers that multiply = “c” When they are added together = “b” from the equation, ax2+bx+c=0
4. Factoring cont. Lets try factoring the equation: x2+5x+6=0 Figure out factors of 6: 5-1, 3-2 Since 3 and 2 add up to 5, they are the factors we want to use (x+3)(x+2)=0
5. Solving Now we have to solve for x from (x+3)(x+2)=0 Set (x+3)=0 and (x+2)=0 Use algebra to solve for x x=-3 and x=2 Solutions of x2+5x+6=0 are -3 and 2.
6. Factoring Harder Equations Try factoring 2x2+x-6=0 Since a≠1, we need factors of a∙c, not just c. a=2 c=-6 2∙-6=-12 Factors of 12: 12-1, 6-2, 3-4 (higher number will be negative) 4 and -3 work since when added equal 1.
7. Factoring cont. Use box method to help factor Factor the rows and columns Upper left= 1st term Lower right=last term Complete the square with the factors (4 and -3) 2x factors out of the first column -3 factors out of second column X factors out of the first row 2 factors out of the second row
8. Solving Now solve for x x+2=0 2x-3=0 x=-2 x=3/2 Our factors from 2x2 +x-6=0 are (x+2)(2x-3)
9. The Quadratic Formula If the equation can’t be factored, we will use the Quadratic Formula From equation ax2+bx+c=0 You should always get two answers
10. Using the Quadratic Formula Solve for x: 2x2-4x-3=0 First identify a, b, and c. a=2 b=-4 c=-3 Plug the numbers into the equation x=-(-4)±√((-4)2-4(2)(-3)) 2(2)
11. Using Formula cont. Now simplify to solve for x x=4±√(16+24) 4 x=4±√(40) 4 Final answer: x=2±√(10) 2
12. More Practice Solve for x: x2-2x-4=0 Identify a, b, and c a=1 b=-2 c=-4 Plug the numbers into the equation x=-(-2)±√((-2)2-4(1)(-3)) 2(1)
13. More Practice cont. Now simplify the equation x=2±√(4+16) 2 x=2±√(20) 2 x=2±2√(5) 2 Final answer: x=1±√(5)
14. The Discriminant The equation under the square root is the Discriminant: b2-4ac This will tell number of roots and if the roots are real or imaginary
15. Discriminant cont. If the Discriminant is negative, there are two real roots. If the Discriminant is zero, there is one real root. If the Discriminant is negative, there are two imaginary roots.
16. Using the Discriminant Use the Discriminant to figure out how many roots 9x2+12x+4=0 has and the type. a=9 b=12 c=4 b2-4ac=(12)2-4(9)(4)=144-144=0 Therefore, there is one real root.
17. Using the Discriminant cont. Use the Discriminant to figure out how many roots 3x2+4x-12=0 has and the type. a=3 b=4 c=-12 b2-4ac=(4)2-4(3)(-12)=16+144=160 Therefore, there are two real roots.
18. Citations Stapel, Elizabeth. "Factoring Quadratics: The Hard Case." Purplemath. Available from http://www.purplemath.com/modules/factquad2.htm Accessed 16 February 2011 Stapel, Elizabeth. "Factoring Quadratics: The Simple Case." Purplemath. Available from http://www.purplemath.com/modules/factquad.htm Accessed 16 February 2011 Quadratic Formula picture. “How to write in quadratic equation.” Blogspot. http://2.bp.blogspot.com/_V8KsSIiGjBk/SsACMEj73KI/AAAAAAAAFIU/vNtErLdchMw/s1600/Quadratic%2BFormula.gif Accessed February 27 2011
