Section 1.6: Solving Other Types of Equations Solving by Factoring – Equations with Higher Degree Polynomials Example: 6 𝑥 4 −14 𝑥 2 =0 2 𝑥 2 3 𝑥 2 −7 =0 2 𝑥 2 =0 3 𝑥 2 −7=0 𝑥 2 =0 3 𝑥 2 =7 𝑥=0 𝑥 2 = 7 3 𝑥=± 7/3

36 𝑥 3 −100𝑥=0 4𝑥 9 𝑥 2 −25 =0 4𝑥 3𝑥−5 3𝑥+5 =0 4𝑥=0 3𝑥−5=0 3𝑥+5=0 𝑥=0 3𝑥=5 3𝑥=−5 𝑥= 5 3 𝑥=− 5 3

𝑥 4 −81=0 𝑥 2 −9 𝑥 2 +9 =0 𝑥−3 𝑥+3 𝑥 2 +9 =0 Remember: You cannot factor the SUM of two squares 𝑥−3=0 𝑥+3=0 𝑥 2 +9=0 𝑥=3 𝑥=−3 𝑥 2 =−9 𝑥=± −9 𝑥=±3𝑖

𝑥 3 +216=0 This is the sum of two cubes: 𝑥 3 +216= 𝑥 3 + 6 3 Factor: 𝑥+6 𝑥 2 −6𝑥+36 =0 𝑥+6=0 𝑥 2 −6𝑥+36=0 𝑥=−6 Use Quadratic Formula: 𝑥= 6± −6 2 −4(1)(36) 2(1) = 6± −108 2 = 6±6 3 𝑖 2 =3±3 3 𝑖

Solve by Factoring 9 𝑥 4 −24 𝑥 3 +16 𝑥 2 =0 𝑥 2 9 𝑥 2 −24𝑥+16 =0 Use Slide and Divide to Factor 9 𝑥 2 −24𝑥+16 𝑥 2 3𝑥−4 3𝑥−4 =0 𝑥 2 =0 3𝑥−4=0 𝑥=0 𝑥= 4 3

Factor by Grouping to Solve 𝑥 3 −3 𝑥 2 −𝑥+3=0 𝑥 2 𝑥−3 −1 𝑥−3 =0 𝑥 2 −1 𝑥−3 =0 𝑥−1 𝑥+1 𝑥−3 =0 𝑥−1=0 𝑥+1=0 𝑥−3=0 𝑥=1 𝑥=−1 𝑥=3

Factor by Grouping to Solve 𝑥 4 +2 𝑥 3 −8𝑥−16=0 𝑥 3 𝑥+2 −8 𝑥+2 =0 𝑥 3 −8 𝑥+2 =0 𝑥−2 𝑥 2 +2𝑥+4 (𝑥+2)=0 𝑥−2=0 𝑥 2 +2𝑥+4=0 𝑥+2=0 𝑥=2 ↓ 𝑥=−2 𝑥= −2± 2 2 −4(1)(4) 2(1) = −2± −12 2 = −2±2 3 𝑖 2 =−1± 3 𝑖

Equations that are Quadratic in Form 𝑥 4 +5 𝑥 2 −36=0 𝑥 2 +9 𝑥 2 −4 =0 𝑥 2 +9 𝑥−2 𝑥+2 =0 𝑥 2 +9=0 𝑥−2=0 𝑥+2=0 𝑥 2 =−9 𝑥=2 𝑥=−2 𝑥=± −9 𝑥=±3𝑖

Equations that are Quadratic in Form 36 𝑥 4 +29 𝑥 2 −7=0 Slide and Divide: 𝑥 4 +29 𝑥 2 −252=0 𝑥 2 +36 𝑥 2 −7 =0 𝑥 2 + 36 36 𝑥 2 − 7 36 =0 𝑥 2 +1 36 𝑥 2 −7 =0 𝑥 2 +1=0 36 𝑥 2 −7=0 𝑥 2 =−1 36𝑥 2 =7 𝑥=± −1 𝑥 2 =7/36 𝑥=±𝑖 𝑥=± 7 /6

Equations that are Quadratic in Form 𝑥 6 +7 𝑥 3 −8=0 𝑥 3 +8 𝑥 3 −1 =0 𝑥+2 𝑥 2 −2𝑥+4 𝑥−1 𝑥 2 +𝑥+1 =0 Solutions: 𝑥=−2 𝑥=1± 3 𝑖 𝑥=1 𝑥= −1± 3 𝑖 2

Equations that are Quadratic in Form 2 𝑥 2/3 +3 𝑥 1/3 −5=0 Slide and Divide: 𝑥 2/3 + 3 𝑥 1/3 −10=0 ( 𝑥 1/3 +5) 𝑥 1/3 −2 =0 𝑥 1/3 + 5 2 𝑥 1/3 − 2 2 =0 ( 2𝑥 1/3 +5) 𝑥 1/3 −1 =0 2𝑥 1/3 +5=0 𝑥 1/3 −1=0 𝑥 1/3 =−5/2 𝑥 1/3 =1 3 𝑥 =−5/2 3 𝑥 =1 𝑥= − 5 2 3 =− 125 8 𝑥= 1 3 =1

Solving Radical Equations: Fraction Exponents 𝑥−5 3/2 =8 𝑥−5 3 =8 Square both sides: 𝑥−5 3 =64 Cube root both sides: 𝑥−5= 3 64 𝑥−5=4 𝑥=9 Shortcut: Take both sides of the original equation to the 2/3 power.

Solving Radical Equations: Fraction Exponents 𝑥 2 −5 2/3 =16 Take both sides to the power of 3/2. [ 𝑥 2 −5 2/3 ] 3/2 = 16 3/2 𝑥 2 −5=64 𝑥 2 =69 𝑥=± 69

More Solving Radical Equations 3𝑥 −12=0 Get the radical expression by itself: 3𝑥 =12 Square both sides: 3𝑥=144 𝑥= 144 3

7 𝑥 −4=0 Get radical expression by itself: 7 𝑥 =4 Square both sides: 7 𝑥 2 = 4 2 49𝑥=16 𝑥= 16 49

Solving Radical Equations with roots other than Square Roots: 3 2𝑥+5 +3=0 Get radical by itself. 3 2𝑥+5 =−3 Cube both sides. 2𝑥+5=−27 2𝑥=−32 𝑥=−16 4 3𝑥+1 −5=0 Get radical by itself. 4 3𝑥+1 =5 Take both sides to 4th power. 3𝑥+1=625 3𝑥=624 𝑥=208

𝑥+ 31−9𝑥 =5 31−9𝑥 =5−𝑥 31−9𝑥 2 = 5−𝑥 2 31−9𝑥= 5−𝑥 5−𝑥 31−9𝑥=25−10𝑥+ 𝑥 2 Put the quadratic equation in general form: 0= 𝑥 2 −𝑥−6 0= 𝑥−3 𝑥+2 𝑥−3=0 𝑥+2=0 𝑥=3 𝑥=−2

− 26−11𝑥 +4=𝑥 − 26−11𝑥 =𝑥−4 − 26−11𝑥 2 = 𝑥−4 2 26−11𝑥= 𝑥−4 𝑥−4 26−11𝑥= 𝑥 2 −8𝑥+16 Put the quadratic equation in general form: 0= 𝑥 2 +3𝑥−10 0= 𝑥+5 𝑥−2 𝑥+5=0 𝑥−2=0 𝑥=−5 𝑥=2

Equations with Two Radical Expressions 𝑥+1 = 3𝑥+1 Square both Sides: 𝑥+1=3𝑥+1 0=2𝑥 0=𝑥

𝑥 − 𝑥−5 =1 Get one radical expression by itself. Then square both sides. 𝑥 = 𝑥−5 +1 𝑥 2 = 𝑥−5 +1 2 𝑥= 𝑥−5 +1 𝑥−5 +1 FOIL: 𝑥= 𝑥−5 +1 𝑥−5 +1 𝑥−5 +1 𝑥=2 𝑥−5 +𝑥−4 Continued on next slide…

𝑥=2 𝑥−5 +𝑥−4 0=2 𝑥−5 −4 Get remaining radical expression by itself: 4=2 𝑥−5 Square both sides: 4 2 = 2 𝑥−5 2 16=4 𝑥−5 16=4𝑥−20 36=4𝑥 Solution: 9=𝑥

𝑥+5 + 𝑥−5 =10 Get one radical expression by itself. Then square both sides. 𝑥+5 =10− 𝑥−5 𝑥+5 2 = 10− 𝑥−5 2 𝑥+5= 10− 𝑥−5 10− 𝑥−5 FOIL: 𝑥+5=100−10 𝑥−5 −10 𝑥−5 +𝑥−5 𝑥+5=95−20 𝑥−5 +𝑥 Continued on next slide…

𝑥+5=95−20 𝑥−5 +𝑥 0=90−20 𝑥−5 Get remaining radical expression by itself: 20 𝑥−5 =90 Square both sides: 20 𝑥−5 2 = 90 2 400(𝑥−5)=8100 400𝑥−2000=8100 400𝑥=10100 Solution: 𝑥= 10100 400 = 101 4

4 𝑥−3 − 6𝑥−17 =3 Get one radical expression by itself. Then square both sides. 4 𝑥−3 =3+ 6𝑥−17 4 𝑥−3 2 = 3+ 6𝑥−17 2 16(𝑥−3)= 3+ 6𝑥−17 3+ 6𝑥−17 FOIL: 16𝑥−48=9+3 6𝑥−17 +3 6𝑥−17 +6𝑥−17 16𝑥−48=−8+6 6𝑥−17 +6𝑥 Continued on next slide…

16𝑥−48=−8+6 6𝑥−17 +6𝑥 Get remaining radical expression by itself: 10𝑥−40=6 6𝑥−17 Square both sides: 10𝑥−40 2 = 6 6𝑥−17 2 10𝑥−40 10𝑥−40 =36(6𝑥−17) 100 𝑥 2 −800𝑥+1600=216𝑥−612 100 𝑥 2 −1016𝑥+2212=0 Use Quadratic Formula to Solve Solution: 𝑥= 1016±384 200 = 1400 200 or 632 200 =7 or 3.16 3.16 is actually an extraneous solution.

Absolute Value Equations Steps for Solving Absolute Value Equations: Get the absolute value expression by itself on one side of the equation. Drop the absolute value. Replace with a ± sign in front of the other side of the equation. Set up two separate equations and solve both of them. Absolute value equations typically have two or more solutions. See examples on the next few slides…

2𝑥−5 −11=0 Get the absolute value expression by itself. 2𝑥−5 =11 2𝑥−5=±11 2𝑥−5=11 2𝑥−5=−11 2𝑥=16 2𝑥=−6 𝑥=8 𝑥=−3

3𝑥+2 =7 3𝑥+2=±7 3𝑥+2=7 3𝑥+2=−7 3𝑥=5 3𝑥=−9 𝑥= 5 3 𝑥=−3

𝑥 2 +6𝑥 =3𝑥+18 𝑥 2 +6𝑥=±(3𝑥+18) 𝑥 2 +6𝑥=3𝑥+18 𝑥 2 +6𝑥=−3𝑥−18 Put in general form: 𝑥 2 +3𝑥−18=0 𝑥 2 +9𝑥+18=0 𝑥+6 𝑥−3 =0 𝑥+6 𝑥+3 =0 Set each factor equal to zero and solve: 𝑥=−6 , 𝑥=3 𝑥=−6 , 𝑥=−3

𝑥−15 = 𝑥 2 −15𝑥 𝑥−15=±( 𝑥 2 −15𝑥) 𝑥−15= 𝑥 2 −15𝑥 𝑥−15=− 𝑥 2 +15𝑥 Put in general form: 0= 𝑥 2 −16𝑥+15 𝑥 2 −14𝑥−15=0 0=(𝑥−15)(𝑥−1) 𝑥−15 𝑥+1 =0 Set each factor equal to zero and solve: 𝑥=15 , 𝑥=1 𝑥=15 , 𝑥=−1 𝑥=1 is actually an extraneous solution.