Algebra 2/Trig Midterm review. Solve and graph equations and inequalities Radical equations:

Algebra 2/Trig

Midterm review

Solve and graph equations and inequalities

• Radical equations:

x =2+ x−2

Solve and graph equations and inequalities

• Radical equations: x =2+ x−2

x−2 = x−2

(x−2)2 =x−2

x2 −4x+ 4 =x−2

x2 −5x+6 =0(x−2)(x−3)=0x=2,3

Absolute value

• Solve and check:

2 x−7 +4 ≥202 x−7 =18

Absolute value

• Solve and check:

2 x−7 +4 ≥20

x−7 ≥8

x−7 ≥8 or x−7 ≤−8

x≥15 or x≤−1

2 x−7 =18

x−7 =9

x−7 =9, x−7 =−9

x=16, −2

Quadratic equations

• Solve: a2 −4a=−6

x2 + x≥12

2x3 −200x=0

Quadratic equations

• Solve:a2 −4a=−6

a2 −4a+6 =0

x=4 ± 16 −4(1)(6)

2(1)

x=4 ± −8

2

x=4 ±2i 2

2=2±i 2

#2

• Set = o:

-4 3

(−∞,−4)∪(3,∞)

# 3

• Factor:2x3 −200x=0

2x(x2 −100)=02x(x+10)(x−10)=0x=0,−10,10

quadratics

• Complete the square to solve: 3x2+6x-45=0

• When will a ball hit the ground, where will it be after 5 seconds what will it’s max height be?

• h(t) = -2t2+40t+4 t is in seconds

Answers:

• Divide out the 3: 3x2+6x-45=0• X2 +2x -15 =0• X2 + 2x + 1 = 15 + 1• (x + 1)2 = 16• X+ 1 = 4 and x + 1 = -4• X = 3 x =-5

Graph:• When will a ball hit the ground, where will it be

after 5 seconds what will is max height be,given• h(t) = -2t2+40t+4

19.05 .94

When t = 20, it hits the ground. After 5 seconds it is 154 ft.high and it reaches a maxheight of 204 ft.

Rational expressions and equations

• Simplify:a2 −6424 −3a

25−x2

4x2 −10x−6•

2x+12x+10

Rational expressions• Simplify: a2 −64

24 −3a=

(a+8)(a−8)3(8 −a)

=−(a+8)

3

-1

Solution:

• Second one:25−x2

4x2 −10x−6•

2x+12x−10

(5 −x)(5 + x)2(2x2 −5x−3)

•2x+1

2(x−5)(5 −x)(5 + x)

2(2x+1)(x−3)•

2x+12(x−5)

5 + x−4(x−3)

-1

Complex Fractions

• Simplify:x−x−1

x−1 −1

Complex Fractions

• Simplify: x • x−1x• x

1x• x −1• x

x2 −11−x

(x+1)(x−1)1−x

−(x+1)

-1

adding

• Find the lcd and add:

2

x +1+

3x2 −1

adding

• lcd = (x+1)(x-1)

2

x +1+ 3

x2 −12(x−1)

(x+1)(x−1)+ 3

x2 −12x−2+3

(x+1)(x−1)2x+1x2 −1

Rational equations

• Multiply by lcd and solve:

a

a−3+

6a(a−3)

=5

a−3

Rational equations

Lcd = a(a-3)

3 is extraneous

a

a−3+

6a(a−3)

=5

a−3a(a(a−3))

a−3+

6(a(a−3))a(a−3)

=5(a(a−3))

a−3

a2 +6 =5a

a2 −5a+6 =0(a−2)(a−3) =0a=2,3a=2

grouping

• Factor and simplify:

x3 −2x2 + 3x−6x−5

•25 −x2

x+5

grouping

• Factor and simplify:x3 −2x2 + 3x−6

x−5•

25 −x2

x2 + 3x−10(x2 + 3)(x−2)

x−5•(5 −x)(5 + x)(x+5)(x−2)

−(x2 + 3)

-1

x2 (x−2)+ 3(x−2)

Functions

• Domain- left to right – x values• Range – bottom to top – y values• Restricted domains:• Set denominators = to 0• Set radicands• f -1 (x) inverse: swap x & y and solve• Varies inversely: xy = xy

≥0

Domain and range:

• Find the largest range for:• Y = 3x – 7• For the domain: −2 ≥ x ≥ 3

Domain and range:

• Find the largest range for:• Y = 3x – 7• For the domain: −2 ≥ x ≥ 3

When x = 3, y = 3(3) – 7 =2Which is the largest value for that domain

Examples:

• Find the domain:

1. f (x)=1

x+ 3

2. f(x) = x2 −2x−15

Examples:

• Find the domain:

• Because it is a denominator and a radical

1. f (x)=1

x+ 3x+ 3> 0x>−3

Examples:• Find the domain: above x axis:

f (x)= x2 −2x−15

x2 −2x−15 ≥0(x−5)(x+ 3)≥0

-3 5

x ≤−3 or x≥5

Compositions:

• Second function inside first:• Let f(x) = x2 + 1 g(x) = x - 3

g o f (x)=

f(g(3)) =

Compositions:

• Second function inside first:• Let f(x) = x2 + 1 g(x) = x - 3

g o f (x)=( )−3

(x2 +1)−3

x2 −2

f(g(3)) =3−3=0

02 +11

Inverses:

f (x)=23

x−43

find f−1(x)

Inverses:f (x)=

23

x−43

find f−1(x)

y=23

x−43

x=23

y−43

x+43

=23

y

32(x+

43

=23

y)

32

x+2 =y or f−1 =32

x+2

Multiply each side by the reciprocal

Irrationals

• Simplify:40a5

27a73

16a11b33

Irrationals

• Simplify:40a5

4 • 10 • a4a

2a2 10a

27a73

3 a6a3

3a2 a3

16a11b33

8∗2 • a9 • a2 • b33

2a3b 2a23

rationalizing

• Rationalize using conjugates:

2+ 31− 3

42 + i

rationalizing

• Rationalize using conjugates:

2+ 31− 3

•1+ 31+ 3

=2+2 3 + 3 + 3

1−3=

5+ 3 3−2

42 + i

•2 −i2 −i

=8 −4i4 −i2

=8 −4i

5

Complex numbers:

• Remember:

Examples:

• Evaluate:2 −9 +3 −4

5 −20 − −5

i31 + i10 + i16 −2i3

Examples:

• Evaluate:

2 −9 +3 −4 =2 • 3i + 3• 2i =6i +6i =12i

5 −20 − −5 =5 • 2i 5 −i 5 =10i 5 −1i 5 =9i 5

i31 + i10 + i16 −2i3 =i3 + i2 +1−2(−i) =−i −1+1+2i =i

The discriminant

• If b2 – 4ac is….• < 0 (negative) roots are IMAGINARY• = 0 roots are rational and equal• > 0, perfect square, roots: rational & unequal • > 0, not perf. Sq., roots: irrational & unequal

• Ex: The roots of ax2 + 12x = -9 are rational and equal when a = ?

Answer:

• ax2 + 12x = -9 • ax2 + 12x + 9 =0• Set b2 – 4ac = 0• 122 – 4(a)(9)=0• 144-36a=0• 36a=144• a=4

Formulas:

• Quadratics:

• x=

• Sum = -b/a• Product = c/a•

−b ± b2 − 4ac

2a

Conjugate roots:

• If 3 – 2i is a root, so is 3 + 2i• Find the equation that has the root 4 – i• STEPS:• 1. Find the sum & the product of 4 – i and its

conjugate• 2. use x2 – sum(x) + product = 0

SOLUTION:

• SUM = PRODUCT = 4 + i4 −i8

(4 + i)(4 −i)

16 −i2

17

x2 −8x+17 =0

circles

• Write the equation of a circle with center at • (-2,3) and a point on the circle (1,1)

• Graph the circle, find the radius using pythagorean theorem and use equation above.

(x−h)2 +(y−k)2 =r2

answer

• Radius =

• Center = (-2,3)

32 +22 = 13

(x +2)2 +(y−3)2 =13

Complete the square for circles

• Example:

x2 + y2 +6x−6y+6 =0

Complete the square for circles

• Example:

x2 +6x+ _9__+ y2 −6y+ _9__ =−6+9+9

(x+ 3)2 +(y−3)2 =12

x2 + y2 +6x−6y+6 =0

Solving exponential equations

• Find like bases and set the exponents equal:

2x+2 =18

Solving exponential equations

• Find like bases and set the exponents equal:

2x+2 =18

2x+2 =123

2x+2 =2−3

x+2 =−3x=−5