Algebra Problems: Simplifying, Solving, and Graphing

This algebra assignment presents solutions to a variety of problems covering fundamental concepts. The assignment begins with simplifying radical expressions and solving radical equations, followed by finding the domain of functions and solving exponential equations. It also includes problems on exponents, logarithms, rational expressions, and systems of equations. Additionally, the assignment covers graphing quadratic functions, determining intercepts, vertex, and solving for x-axis interception points. Finally, the assignment concludes with graphing piecewise functions and solving miscellaneous algebra problems. The solutions are detailed and step-by-step, offering a comprehensive guide to understanding and solving the presented algebraic challenges. This resource is designed to assist students in grasping key algebraic concepts and improving their problem-solving skills.

Algebra Problems

Q1. Simplify the questions??
a). 2√128
Ans: √{27}
√{26.2}
23 √{2}
=8 √{2}
b). √{9 a8}b3
Ans: √{9}√{a8}
3√{a8}
= 3a {8}/{2}}
= 3 a4 b3
C). 3√{24 a4}b8
Ans: 2[3]√3 b8 a{{4}/{3}}
d). √{{75}/{a6}}
Ans: √{52. 3}
√{3}√{52}
{5√{3}}/{a3}
e) {5}/√{7}}
Ans: frac5√{7}√{7}√{7}}
{5√{7}}/{7}
f).{3 y}√{4-{7}}
Ans: {3 y}/{4-√{7}}={(4+√{7})y}/{3}
3 y(4+√7)/(4-√7)(4+√7) : Multiply by the conjugate
(4-V7)(4+V7)= 9
= 3(4+V7)y/9
= (4+V7)y/3

Q2. Solve following radical equations??
a). √{7-5x}=8
Ans: (√{7-5x})2=82
7-5x=64
x=-{57}/{5}
b) √{2y+15}=√{4y+1}
Ans: √{2y+15})^2=(√{4y+1})2
2y+15=4y+1
=> y=7
c). [3]√{3x+4+2}=0
Ans : [3]√{3x+4+2}3=03
3x+6=0
x=- 2
d). (x-3)2/3=4
Ans: 3(x-3)2/3 =4. 3
2(x-3)=12
frac{2(x-3}{2}={12}/{2}
x=9
Q3 Find Domain of following function??
a). ∫(x)= x{2}+4
Ans: ∫ xdx
frac{x{1+1}}{1+1}
= frac{x2}{2}
= frac{x2}{2}+C
b). ∫(x)= {1}/{x}+{2}/{x+2}

Ans: ∫xdx
{x{1+1}}/{1+1}
= {x2}/{2}+C
c). g(x)= {x}/(x2-5 X)}
Ans: Domain of {x}/(x2-5 X)} [ Sol: x<0 or 0<x<5 or x>5]
Asymptotes of {x}/(x2-5 X)} : Vertical x=0, x=5, Horizontal: y=0
Extreme Points of {x}/(x2-5 X)} : None
d). h(x)= √4-x
Ans: √4-x : X intercepts: (4,0), Y intercepts: (0,2)
Extreme Points of √4-x : Minimum(4,0)
e) g(x)= √{x^2}+1
Ans: Asymptotes of √{x^2}+1 : Horizontal y= x+1, y=-X+1
= √{x^2}+1 : minimum (0,1)
Q 4. Solve the exponential equations?
a). ({1}/{2})x/3=1/4
Ans {1}/{2})x/3.4 =1
{1}/{2})x/3.4}/{4}={1}/{4}
= (1/2){f{x}/{3}}=(2^2)^{-1}
= 2-1. {x}/{3}=2^{2(-1)}
= -{x}/{3}=- 2
=> x= 6
b). 3. 4x/2=96
Ans: 3.4x/2=96
4x/2=32
(22)x/2=32
(22)x/2=32

2.{x}/{2}=5
x= 5
c) ex2 +5x =e-6
=x2+5x=-6
Ans: X=-2, x=-3
d) 3(5-x/4)=15
Ans: 3(5-fx/4)=15
= 5^-x/4=5
= -{x/4}=1
= x=-4
3. Direction y= -3 x
IV Exponents?
1. (81/64)1/2
Ans:811/2641/2
(81/8)1/2
= 9/8
2. (27-2)1/3
Ans: (-2){1}/{3}
-2.{1}/{3}
= {2}/{3}
= 272/3
=1/9
3. -24
Ans: -24=-15
= -16
4. (-2)4

Ans: =24
=16
5. (4-1+2-1})2
Ans: {1}/{4}+{1}/{2})2
{3}/{4})2
{3^2}/{42}
={9}/{16}
6. (13y)-1
Ans : {1}/{13y}
7. 13y-1
Ans : 13.{1}/{y}
{1/13}/{y}
= {13}/{y}
8. 8-1. 80
Ans : 8-1.1
= 1.{1}/{8}
= {1}/{8}
9. (3{-5}.3-10}/{(32}
Ans: {0.3-10/37
1/0.310/37
={1}/{0.0129140163}
= 77.43524
10. 3-3.0.3-10/32
Ans: 3-3/32 = 1/32-(-3)
= 0.3-10/ 35

=0.3-10/35 : Apply exponent rule : xa/xb = 1/xb-a
= 1/35. 0.310
= 1/0.0014348907
= 696.91719
V.Logarithms
i. Factor and cancel
1. 3log2x=12
=3log2(x)/3= 12/3 (divide both sides by 3)
=log2(x)= 4
Apply rule a= logb (bª)
4= log2(24)= log2(16)
Log2(x)= log2(16)
x=16
2. log5125=x
x= log5125
x= log5 (53)
Apply rule loga(xb )= b.loga(x)
x= log5(53)= 3log5(5)
x= 3log5(5)
Apply rule: loga(a)=1
x=3*1
x=3
3. 3+4logx4=5
= 3+4logx(4)=5
= 3+4logx(4)-3=5-3 (subtract bith sides by 3)
= 4logx(4)=2
= 4logx(4)/4=2/4 (divide both sides by 4)
= logx(4)= 1/2

Apply rule: loga(b)= In (b)/ In (a)
logx (4)= In (4)/ In (x)
In (4)/ In (x)= 1/2
In (4)*2= In (x)*1
In (x)*1= In (4)*2
In (x)= In (4)*2
(Simplify In (4)*2: 4In (2))
In (x)= 4In (2)
In (x)= In (16)
x=16
Rational Expression
i. Factor and cancel.
1. a3-a2-6a/a2-9
=a(a+2) (a+3)/a2-9 [factor a3-a2-6a: a(a+2) (a+3)]
= a(a+2)/a+3/(a+3) (a-3) [factor a2-9: (a+3) (a-3)]
= a(a=2)/a+3
2. a3-2a2+a-2/2-a
=(a-2)(a2+1)/-a+2 [factor a3-2a2+a-2: (a-2)(a2+1)]
=-(a2+1)
3. z3+8/z2-2z+4
= z3+8/z2-2z+4 [factor z3+8: (z+2)(z2-2z+4 ]
=(z+2)(z2-2z+4/ z2-2z+4)
=(z+2)
4. 4ab2/ab2+a
= 4ab2/a(b2+1)
=4b2/(b2+1)

5. x2+6x+8/x2-4
= (x+2) (x+4)/x2-4 [Factor x2+6x+8: (x+2) (x+4)]
= (x2+2x) + (4x+8) [Break expression into groups]
= x(x+2) + 4(x+2) [Factor (x2+2x): x(x+2)), (Factor (4x+8): 4(x+2)]
= (x+2) (x+4) [common factor: (x+2)]
=(x+2) (x+4)/ x2-4
= (x+2) (x+4)/(x+2) (x-2) [Factor (x2+4): (x+2) (x-2)]
= (x+4)/(x-2)
ii. Perfrom the operation and write the result in reduced form
1. (a-1/a)*( a2/a2-1)
=((a-1)a2)/a( a2-1) [Multiply Fraction a/b* c/d= a*c/b*d]
=(a(a-1))/( a2-1)
=(a(a-1))/( a-1)(a+1) [Factor a2-1: ( a-1)(a+1)]
= a/(a+1)
2. ( a2-9/(a+2))*((a2+4a+4)/ a2-a-6)
= ((a2+4a+4)/ (a2-a-6))= (a+2)/(a-3)
=a2-9/(a+2)*(a+2)/(a-3)
=a2-9/(a+2)/(a+2)/(a-3) [ Multiply Fraction a/b* c/d= a*c/b*d]
= a2-9/(a-3)
= (a+3)(a-3)/(a-3)
=a+3 [Factor a2-9: ( a+3)(a-3)]
3. (x+4/x2)+ (x2-16/x)
=(x+4/x2 )+ ((x2-16)x/x2)

= x+4+(x2-16)x/x2
=x2-15x+4/x2
iii. Get a common denominator
1. (2/2x+1)-( 5/(2x+1)2)
= 2*(2x+1)/(2x+1)2-( 5/(2x+1)2) [ Least Common Multiplier of 2x+1, (2x+1)2:(2x+1)2]
= 2 (2x+1)-5/(2x+1)2
=4x-3/(2x+1)2 [Expand 2 (2x+1)-5: 4x-3]
2. 3- (4/x+2)
= 3 (x+2)/x+2-(4/x+2) [Convert element of fraction: 3= 3 (x+2)/x+2]
=3 (x+2)-4/ x+2
=3x+2/x+2
3. (3/x-2)+(5/2-x)
=3(-x+2)/(x-2)(-x+2) + 5(x-2)/(2-x)(x-2)
=3 (-x+2)+5(x-2)/(x-2)(-x+2)
=2(-x+2)/(x-2)(-x+2)
=-(2/x-2)
4. (y/y2+4y+4)+(3/y2+y-2)
= (y/(y+2)2)+ 3/( y2+y-2)
=(y/(y+2)2)+ 3/( y-1)(y+2)
=y(y-1)+3(y+2)/(y+2)2 (y-1)
= (y2+2y+6)/(y+2)2(y-1)
iv. Get a common denominator in the numerator and multiply by reciprocal, or mutliply by
LCD/LCD.

1. (1-y/3)/ 3-y
=(3-y/3)/3-y
= 3-y/ 3(-y+3)
=1/3
2. ((1/y+1)-1/y)/ (1/y+1)
=((1/y+1)-(1/y) (y+1))/1
=(-(1/y(y+1))(y+1)/1
=-((1/y(y+1)(y+1)/1
=-1/y/1
= -1/y
3. ((x/x-1)+1)/x+2/x
=x((x/x-1)+1)/(x+2)
=(2x-1/x-1)x/(x+2)
=x(2x-1)/(x-1)(x+2)

Q Given y= -3x2+x+2, Find and Graph
a. Y- intercepts
b. Vertex
c. X- intercepts
Ans: 3x2+y= x+2
Y= -(X-1)(3x+2)
3X2- x+y-2= 0
X= -2/3, Y= 1
For parabola ax2+bx+c the vertex x equal -b/2a
a= -3, b=1
x= -1/2(-3)
x=1/6
Putting equation x=1/6
y= -3(1/6)2+ 1/6+2
y= 25/12
Vertex (1/6,25/12)
X- axis interception points of -3x2+x+2 : (-2/3, 0), (1,0)
Y- axis interception point of -3x2+x+2 : (1,0)
Vertex of -3x2+x+2 : Maximum (1/6, 25/12)