Warm-Up Exercises Factor. 1. x 2 2x + 48 – ANSWER ( ) 8 + x 6 – 2. 9y 2 27y 20 + – ANSWER ( ) 4 3y – 5 3. 2y 2 16y 32 + – ANSWER ( ) 4 y – 2 The sides of a square are (2s 3) inches. Find the area. + ANSWER 4s 2 12s 9 + – ( in.2 )

Example 1 Factor the expression. a. m 2 – 25 b. q 2 – 625 c. 9y 2 – 16 Factor a Difference of Two Squares Factor the expression. a. m 2 – 25 b. q 2 – 625 c. 9y 2 – 16 SOLUTION Write as . a. m 2 – 25 = 52 a 2 b 2 Difference of two squares pattern = ( ) 5 + m – Write as . b. = q 2 – 252 a 2 b 2 625 Difference of two squares pattern = ( ) 25 + q – 2

Example 1 c. 9y 2 – 16 = 42 ( )2 3y = ( ) 4 + 3y – Factor a Difference of Two Squares c. 9y 2 – 16 = 42 ( )2 3y Write as . a 2 – b 2 Difference of two squares pattern = ( ) 4 + 3y –

Checkpoint Factor the expression. 1. x 2 – 36 ANSWER ( ) 6 + x – 2. Factor Using the Difference of Two Squares Pattern Factor the expression. 1. x 2 – 36 ANSWER ( ) 6 + x – 2. r 2 – 100 ANSWER ( ) 10 + r – 3. 9m 2 – 64 ANSWER ( ) 8 + 3m – 4. p 2 – 81 4 1 ANSWER p + 2 1 9 –

Example 2 Factor the expression. a. m 2 16m + 64 b. 9p 2 30p + 25 c. Factor a Perfect Square Trinomial Factor the expression. a. m 2 16m + 64 b. 9p 2 30p + 25 c. 16r 2 56r + 49 – SOLUTION m 2 16m + 64 = 2 ( ) m 8 82 a. Write as a 2 2ab b 2 . ( )2 8 m + = Perfect square trinomial pattern

Example 2 b. 9p 2 30p + 25 = ( ) 5 52 )2 3p 2 ( )2 5 3p + = c. 16r 2 Factor a Perfect Square Trinomial b. 9p 2 30p + 25 = ( ) 5 52 )2 3p 2 Write as a 2 2ab b 2 . Perfect square trinomial pattern ( )2 5 3p + = c. 16r 2 56r + 49 – = ( ) 7 72 )2 4r 2 Write as a 2 2ab b 2 . ( )2 7 4r = – Perfect square trinomial pattern 6

[ ] Example 3 Factor . 5u 2 40u + 80 – SOLUTION 5u 2 40u + 80 – = ( ) Factor Out a Common Constant Factor . 5u 2 40u + 80 – SOLUTION 5u 2 40u + 80 – = ( ) u 2 8u 16 5 Factor out 5. ( ) 4 + 42 u 2 2 – u [ ] = 5 Write as a 2 2ab b 2 . = ( )2 4 u – 5 Perfect square trinomial pattern CHECK Check your answer by multiplying. ( ) 4 u – = )2 5 u 2 8u + 16 5u 2 40u 80 7

Checkpoint Factor the expression. 5. x 2 10x + 25 ANSWER ( )2 5 x + 9 6. ANSWER ( )2 3 2x + 7. 9p 2 24p + 16 – ANSWER ( )2 4 3p – 8. 5x 2 10x + 5 ANSWER ( )2 1 x + 5

Checkpoint Factor the expression. 9. 8y 2 18 – ANSWER ( ) 3 2y – + 2 10. 12u 2 36u + 27 – ANSWER ( )2 3 2u –

Example 4 Solve – 9p 2 = 25. + 30p SOLUTION – 9p 2 = 25 + 30p 9p 2 = + Solve a Quadratic Equation Solve – 9p 2 = 25. + 30p SOLUTION Write original equation. – 9p 2 = 25 + 30p Write in standard form. 9p 2 = + 30p 25 ( )2 Write as 3p = + 2 52 ) 5 a 2 2ab b 2. Perfect square trinomial pattern = ( )2 3p + 5 Use the zero product property. = 3p + 5 Solve for p. = p 3 5 – 10

Example 4 Solve a Quadratic Equation ANSWER The solution is . 3 5 – 11

Find the circumference of a rope that can be used to lift 720 pounds. Example 5 Use a Quadratic Equation as a Model Rope Strength Every rope has a safe working load. A rope should not be used to lift a weight greater than its safe working load. The safe working load S (in pounds) for a rope can be found using the function S 180C 2 where C is the circumference (in inches) of the rope. Find the diameter of rope needed to lift an object that weighs 720 pounds. = SOLUTION STEP 1 Find the circumference of a rope that can be used to lift 720 pounds. = 180C 2 S Function for safe working load 12

Example 5 720 = 180C 2 = 180C 2 – 720 = 180 – 4 ( ) C 2 = 180 + 2 ( ) Use a Quadratic Equation as a Model 720 = 180C 2 Substitute 720 for S. = 180C 2 Write in standard form. – 720 = 180 Factor out 180. – 4 ( ) C 2 = 180 Difference of two squares pattern + 2 ( ) C – = + 2 C or – Use the zero product property. = 2 or C Solve for C. – Reject the negative value of C. The rope must have a circumference of 2 inches. 13

Find the diameter of the rope. Use the formula for circumference, C = Example 5 Use a Quadratic Equation as a Model STEP 2 Find the diameter of the rope. Use the formula for circumference, C = πd. Substitute 2 for C in C = πd 2 πd. Solve for d. = d 2 π ≈ 0.637 ANSWER The rope must have a diameter of or about 0.637 inch. 2 π

Checkpoint Solve the equation. 11. x 2 – ANSWER 3 6x = – 9 12. 12 4 Solve a Quadratic Equation Solve the equation. 11. x 2 – ANSWER 3 6x = – 9 12. 12 4 a 2 = – ANSWER 4, 4 – 13. 200 40y 2y 2 = – ANSWER 10