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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 ≥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
a2 −4a+6 =0
x=4 ± 16 −4(1)(6)
2(1)
x=4 ± −8
2
x=4 ±2i 2
2=2±i 2
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.
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
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−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
When x = 3, y = 3(3) – 7 =2Which is the largest value for that domain
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)=( )−3
(x2 +1)−3
x2 −2
f(g(3)) =3−3=0
02 +11
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
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
•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
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
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
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
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