5-4 Factoring Quadratic Expressions

1. 5-4 Factoring Quadratic Expressions Perfect square trinomials

2. What is a perfect square trinomial? • A perfect square trinomial is the product you get when you square a binomial (you multiply the binomial by itself) • To square a binomial • Example: (x+5) (x+5) • square the first terms _____ x _____= _____ • square the last terms _____ x _____= _____ • the middle term is two times the product of the terms 2( _____ _____) = ______ The result is x2 + 10x + 25

3. Factoring a perfect square trinomial when all terms are positive • 9x2 + 48x + 64 • Take the square root of the first and third terms • First term (3x) third term (8) • Put the terms in the parenthesis squared and separate the terms with a plus sign. • (3x + 8)2

4. 4x2 + 24x + 36 • Take the square root of the first and third terms and separate them by an addition sign • (2x + 6)2 • Check your answer: • Square the first term (2x)2 = 4x2 • Square the last term (6)2 = 36 • Multiply the two terms together and double them • (2(2x)(6)) =24x Try this one: 4x2 + 28x + 49

5. Factoring a perfect square trinomial when the middle term is negative • 4n2 – 20n + 25 • The only difference with factoring these trinomials is that the sign between the two square roots is negative • Take the square root of the first and third terms • First term (2n) third term (5) • Separate the terms with a negative sign • (2n - 5)2

6. 4n2 – 16n + 16 • Take the square root of the first and third terms and separate the numbers with a subtraction sign • (2n - 4)2 • Check your answer: • Square the first term (2n)2 = 4n2 • Square the last term (4)2 = 16 • Multiply the two terms together and double them • (2(2n)(-4)) =-16n Try this one: 9x2 - 42x + 49

7. Factoring the difference of two squares • The difference of two squares is written as: a2 – b2 • When they are factored they become (a+b)(a-b) • When the terms of the binomials are FOIL’d the middle terms cancel each other out because one is positive and one is negative • (a+b)(a-b) = a2+ ab – ab - b2 = a2 – b2 • Take the square root of the first term and the square root of the second term and place them into two sets of parentheses – one set separated with a plus sign and one set separated with a minus sign

8. Example: c2 – 64 = (c+8)(c-8) Square root of the first term is c Square root of the last term is 8 Put the terms into two parentheses and separate one with a plus sign and one with a minus sign Remember: the middle terms will cancel out when the binomials of the difference of two squares are multiplied together.

9. Let’s try this one: 4x2 – 16 • Take the square root of the first term: 2x • Take the square root of the last term: 4 • Rewrite the terms in two parentheses • (2x 4) (2x 4) • Separate the terms by a plus sign in one of the parentheses and a minus sign in the other parentheses (2x + 4) (2x – 4) • How about this one: 9x2 – 36

10. One last note: • Sometimes you may have to factor out the GCF before you can factor the quadratic. • You can try to find factors of the first term and then find factors of the last term to make two binomials that you can multiply together • If you can factor out a GCF first – do so in order to make the factoring easier • 3n2 – 24n – 27 • 3(n2 – 8n – 9) • 3(n – 9)(n + 1) • This technique works for any trinomial you are trying to factor

11. h/w: p. 264: 38, 39, 41, 42, 44, 45, 52, 55, 57, 58