At the end of this lesson, students will be able to:
- Write quadratic equations in standard form.
- Factor quadratic expressions for different coefficient values.
- Factor when a=−1.
Terms introduced in this lesson:
Teaching Strategies and Tips
In this lesson, students learn to factor quadratic polynomials according to the signs of a,b, and c:
a=1,b>0,c>0. See Examples 1-4.
a. x2+15x+26. Answer: (x+13)(x+2)
b. x2+13x+40. Answer: (x+8)(x+5)
c. x2+20x+75. Answer: (x+15)(x+5)
a=1,b<0,c>0. See Examples 5 and 6.
a. x2−17x+42. Answer: (x−14)(x−3)
b. x2−21x+90. Answer: (x−15)(x−6)
c. x2−14x+48. Answer: (x−6)(x−8)
a=1,c<0. See Examples 7-9.
a. x2−15x−54. Answer: (x−18)(x+3)
b. x2+7x−60. Answer: (x+12)(x−5)
c. x2−16x−192. Answer: (x−24)(x+8)
a. −x2−4x+60. Answer: −(x−6)(x+10)
b. −x2+14x−40. Answer: −(x−10)(x−4)
c. −x2−25x−156. Answer: −(x+12)(x+13)
- Allow students to infer that if c>0(a=1), then the factorization will be either of the form (−−+−−)(−−+−−) or (−−−−−)(−−−−−) (same signs). If c<0 (a=1), then use the form (−−−−−)(−−−−−) (different signs).
- See summary at the end of the lesson for a list of procedures and examples for each case.
Emphasize that factoring is the reverse of multiplication.
- Use an example such as (x+3)(x+7)=x2+10x+21 in which the binomials are expanded one step at a time to motivate factoring.
- Demonstrate that factoring is equivalent to putting squares and rectangles back together into larger rectangles.
Solution. The diagram shows that (x+3)(x+7)=x2+10x+21. Observe that it also shows how to factor x2+10x+21.
Suggest that students stop listing the possible products for c after the correct choice is evident.
General Tip: For quadratic trinomials with a=−1, remind students to factor −1 from every term. Remind students to include it in their final answer.