9.6 Perfect Squares & Factoring

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Presentation transcript:

9.6 Perfect Squares & Factoring

Perfect Square Trinomials First term must be a perfect square Last term must be a perfect square Middle term must equal 2(first term)(last term) Then, a2+2ab+b2=(a+b)2 and a2-2ab+b2=(a+b)2

Determine whether is a perfect square trinomial. If so, factor it. 1. Is the first term a perfect square? Yes, 2. Is the last term a perfect square? Yes, 3. Is the middle term equal to ? Yes, Answer: is a perfect square trinomial. Write as Factor using the pattern. Example 6-1a

Determine whether is a perfect square trinomial. If so, factor it. 1. Is the first term a perfect square? Yes, 2. Is the last term a perfect square? Yes, 3. Is the middle term equal to ? No, Answer: is not a perfect square trinomial. Example 6-1a

Answer: not a perfect square trinomial Determine whether each trinomial is a perfect square trinomial. If so, factor it. a. b. Answer: not a perfect square trinomial Answer: yes; Example 6-1b

Factor . First check for a GCF. Then, since the polynomial has two terms, check for the difference of squares. 6 is the GCF. and Factor the difference of squares. Answer: Example 6-2a

Factor . This polynomial has three terms that have a GCF of 1. While the first term is a perfect square, the last term is not. Therefore, this is not a perfect square trinomial. This trinomial is in the form Are there two numbers m and n whose product is and whose sum is 8? Yes, the product of 20 and –12 is –240 and their sum is 8. Example 6-2a

Answer: Write the pattern. and Group terms with common factors. Factor out the GCF from each grouping. is the common factor. Answer: Example 6-2a

Factor each polynomial. a. b. Answer: Answer: Example 6-2b

Solving Perfect Square Trinomials

Recognize as a perfect square trinomial. Solve Original equation Recognize as a perfect square trinomial. Factor the perfect square trinomial. Set the repeated factor equal to zero. Solve for x. Answer: Thus, the solution set is Check this solution in the original equation. Example 6-3a

Solve Answer: Example 6-3b

Separate into two equations. or Solve . Original equation Square Root Property Add 7 to each side. Separate into two equations. or Simplify. Answer: The solution set is Check each solution in the original equation. Example 6-4a

Recognize perfect square trinomial. Solve . Original equation Recognize perfect square trinomial. Factor perfect square trinomial. Square Root Property Subtract 6 from each side. Example 6-4a

Separate into two equations. or Separate into two equations. Simplify. Answer: The solution set is Check this solution in the original equation. Example 6-4a

Subtract 9 from each side. Solve . Original equation Square Root Property Subtract 9 from each side. Answer: Since 8 is not a perfect square, the solution set is Using a calculator, the approximate solutions are or about –6.17 and or about –11.83. Example 6-4a

Check You can check your answer using a graphing calculator. Graph and Using the INTERSECT feature of your graphing calculator, find where The check of –6.17 as one of the approximate solutions is shown. Example 6-4a

Solve each equation. Check your solutions. a. b c. Answer: Answer: Answer: Example 6-4b