09-6+Perfect+Squares+and+Factoring

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**__Summary__:** Section 9-6 is about factoring perfect square trinomials, and solving equations involving perfect squares. Numbers like 4, 16, 25 are all perfect squares. To factor a perfect square trinomials, you have to set up an equation. a + b = (a+b) (a+b). For a trinomial to be able to be factored, there are three conditions. The first and last terms must be perfect squares. The term in the middle must be twice the product of the square roots of both the first and third terms. For example, 4x2 + 20x + 25.

**__Rules, Properties, and Formulas__** Perfect Square Trinomials are trinomials are the square of a binomial. So, for example, if you have (x+7)² it can also be written as (x+7)(x+7). You then FOIL it to get x² + 7x + 7x + 49. It is then simplified to x² + 14x +49. This is a perfect square trinomial (PST).

You can tell if a trinomial is a PST by finding the square root of the first and last terms, multiplying them together, and then multiplying by 2. If this is equal to the middle number, it is a PST. Example: c² - 6c + 9. The square root of c² is c, and the squre root of 9 is 3. 3 times c is 3c, times 2 is 6c. This is a PST.  Rules: 1. The first and last terms must be perfect squares. 2. The middle term must be twice the product of the square roots of the first and last terms. 3. Only the middle term can be negative.  //Steps for Factoring:// 1. Identify the trinomial as a PST. 2. Take the square root of the first term and the third term and put them in parenthesis, squared. The sign in between them is dictated by the middle term.
 * if there is no GCF and no simplifying can be done and the problem does not fit all of these rules, then it is prime.

Formula with variables: Example 1: a² + 2ab + b² (a+b)²

Example 2: a² - 2ab + b² (a-b)²

Examples with numbers: Example 1: r² + 4r + 4 (r+2)²

Example 2: g² - 14g + 49 (g - 7)²

Example 3: 25n² - 10x - 1 Prime

Example 4: 36n² + 48n + 64 4(9n² + 12n + 16) 4(3n + 4)²

//Steps for Solving//: 1. Identify as a PST. 2. Factor (see step 2 under //Steps for Factoring//). 3. Since you have basically 2 of the same equations, you only need to set it equal to zero once. 4. Solve the mini equation. 

Example 1: 81 + 18x + x² (x+9)² x+9=0 x = -9

Important Links : [|Algebra help link]

[|E-how.com link]

For more instructions and examples, here's a video: []