Practice Problem Collection

Algebra Practice Problems

60 original problems with detailed solutions, LaTeX syntax, and MATLAB verification
How to use

Work each problem on paper first. Click Show Solution only after you have attempted it. Each solution shows the full worked process, the LaTeX code, and a MATLAB command to verify.

Exponents & Radicals
Problem 1

Simplify: $\displaystyle \frac{4x^{-3}y^4}{8x^2 y^{-2}}$

Solution

Coefficients: $4/8 = 1/2$. Quotient rule: $x^{-3-2}=x^{-5}$, $y^{4-(-2)}=y^6$.

$$\boxed{\frac{y^6}{2x^5}}$$

LaTeX: \frac{y^6}{2x^5}

syms x y; simplify(4*x^(-3)*y^4 / (8*x^2*y^(-2)))
Problem 2

Simplify: $(3a^4 b^{-3})^{-2}$

Solution

Distribute $-2$: $3^{-2}a^{-8}b^{6}$.

$$\boxed{\frac{b^6}{9a^8}}$$

LaTeX: \frac{b^6}{9a^8}

syms a b; simplify((3*a^4*b^(-3))^(-2))
Problem 3

Evaluate: $64^{2/3}$

Solution

Denom 3 = cube root, numer 2 = power: $(\sqrt[3]{64})^2 = 4^2 = 16$.

$$\boxed{16}$$

LaTeX: 64^{2/3} = \left(\sqrt[3]{64}\right)^2 = 16

64^(2/3)  % 16
Problem 4

Simplify: $\displaystyle \frac{a^{5/6} \cdot a^{1/3}}{a^{1/2}}$

Solution

Common denom 6: $a^{5/6+2/6-3/6}=a^{4/6}=a^{2/3}$.

$$\boxed{a^{2/3}}$$
syms a; simplify(a^(5/6)*a^(1/3)/a^(1/2))
Problem 5

Simplify: $\sqrt{98}+3\sqrt{2}-\sqrt{50}$

Solution

$\sqrt{98}=7\sqrt{2}$, $\sqrt{50}=5\sqrt{2}$: $7\sqrt{2}+3\sqrt{2}-5\sqrt{2}=5\sqrt{2}$.

$$\boxed{5\sqrt{2}}$$
syms x; simplify(sqrt(98)+3*sqrt(2)-sqrt(50))
Problem 6

Rationalize: $\displaystyle \frac{6}{5-\sqrt{7}}$

Solution

Multiply by conjugate: $\frac{6(5+\sqrt{7})}{25-7}=\frac{6(5+\sqrt{7})}{18}$.

$$\boxed{\frac{5+\sqrt{7}}{3}}$$
syms x; simplify(6/(5-sqrt(7)))
Polynomials & Factoring
Problem 7

Expand: $(4x+1)(2x^2-3x+5)$

Solution

Distribute each term: $8x^3-12x^2+20x+2x^2-3x+5$.

$$\boxed{8x^3-10x^2+17x+5}$$
syms x; expand((4*x+1)*(2*x^2-3*x+5))
Problem 8

Factor completely: $3x^3-75x$

Solution

GCF then difference of squares: $3x(x^2-25)$.

$$\boxed{3x(x+5)(x-5)}$$
syms x; factor(3*x^3-75*x)
Problem 9

Factor: $8x^3-27$

Solution

Difference of cubes with $a=2x$, $b=3$.

$$\boxed{(2x-3)(4x^2+6x+9)}$$
syms x; factor(8*x^3-27)
Problem 10

Factor: $10x^2-11x-6$

Solution

$ac=-60$. Numbers: $4$ and $-15$. Group: $2x(5x+2)-3(5x+2)$.

$$\boxed{(2x-3)(5x+2)}$$
syms x; factor(10*x^2-11*x-6)
Problem 11

Factor completely: $x^4-81$

Solution

$(x^2+9)(x^2-9)=(x^2+9)(x+3)(x-3)$. $x^2+9$ is irreducible over $\mathbb{R}$.

$$\boxed{(x^2+9)(x+3)(x-3)}$$
syms x; factor(x^4-81)
Rational Expressions & Complex Numbers
Problem 12

Simplify: $\displaystyle\frac{x^2-16}{x^2+2x-8}$

Solution

$\frac{(x+4)(x-4)}{(x+4)(x-2)}=\frac{x-4}{x-2}$, $x\neq -4$.

$$\boxed{\frac{x-4}{x-2}}$$
syms x; simplify((x^2-16)/(x^2+2*x-8))
Problem 13

Add: $\displaystyle\frac{3}{x+2}+\frac{5}{x-3}$

Solution

LCD=$(x+2)(x-3)$: $\frac{3(x-3)+5(x+2)}{(x+2)(x-3)}=\frac{8x+1}{(x+2)(x-3)}$.

$$\boxed{\frac{8x+1}{(x+2)(x-3)}}$$
syms x; simplify(3/(x+2)+5/(x-3))
Problem 14

Compute: $(5-3i)(2+7i)$

Solution

FOIL: $10+35i-6i-21i^2=10+29i+21$.

$$\boxed{31+29i}$$
(5-3i)*(2+7i)  % 31+29i
Problem 15

Divide: $\displaystyle\frac{4-i}{2+3i}$

Solution

Conjugate: $\frac{(4-i)(2-3i)}{(2+3i)(2-3i)}=\frac{8-12i-2i+3i^2}{4+9}=\frac{5-14i}{13}$.

$$\boxed{\frac{5}{13}-\frac{14}{13}i}$$
(4-1i)/(2+3i)
Solving Equations
Problem 16

Solve: $\displaystyle\frac{x}{4}-\frac{x+1}{6}=2$

Solution

LCD=12: $3x-2(x+1)=24$, so $x-2=24$.

$$\boxed{x=26}$$
syms x; solve(x/4-(x+1)/6==2,x)
Problem 17

Solve for $h$: $V=\frac{1}{3}\pi r^2 h$

Solution
$$\boxed{h=\frac{3V}{\pi r^2}}$$
syms V r h; solve(V==pi*r^2*h/3,h)
Problem 18

Solve by factoring: $x^2+3x-28=0$

Solution

$(x+7)(x-4)=0$.

$$\boxed{x=-7\text{ or }x=4}$$
syms x; solve(x^2+3*x-28==0,x)
Problem 19

Solve: $5x^2-3x-1=0$

Solution

$D=9+20=29$.

$$\boxed{x=\frac{3\pm\sqrt{29}}{10}}$$

LaTeX: x=\frac{3\pm\sqrt{29}}{10}

syms x; solve(5*x^2-3*x-1==0,x)
Problem 20

Complete the square: $x^2-10x+7=0$

Solution

$(x-5)^2-25+7=0$, so $(x-5)^2=18$.

$$\boxed{x=5\pm 3\sqrt{2}}$$
syms x; solve(x^2-10*x+7==0,x)
Problem 21

Discriminant only: $4x^2+4x+1=0$

Solution

$D=16-16=0$. One repeated real root: $x=-\frac{1}{2}$.

$$\boxed{D=0\text{ — one repeated root}}$$
a=4;b=4;c=1; D=b^2-4*a*c  % 0
Problem 22

Solve: $\sqrt{3x+4}=x+2$

Solution

Square: $3x+4=x^2+4x+4$, so $x^2+x=0$, $x(x+1)=0$. Check both: $x=0$ ✓, $x=-1$ ✓.

$$\boxed{x=0\text{ or }x=-1}$$
syms x; solve(sqrt(3*x+4)==x+2,x)
Problem 23

Solve: $x^4-10x^2+9=0$

Solution

Let $u=x^2$: $(u-1)(u-9)=0$, so $u=1,9$.

$$\boxed{x=\pm 1,\;\pm 3}$$
syms x; solve(x^4-10*x^2+9==0,x)
Inequalities & Absolute Value
Problem 24

Solve: $-5(x-2)\geq 15$

Solution

$-5x+10\geq 15$, $-5x\geq 5$. Divide by $-5$ (flip!): $x\leq -1$.

$$\boxed{(-\infty,-1]}$$
syms x; solve(-5*(x-2)>=15,x,'ReturnConditions',true)
Problem 25

Solve: $x^2+2x-15>0$

Solution

$(x+5)(x-3)>0$. Test intervals.

$$\boxed{(-\infty,-5)\cup(3,\infty)}$$
syms x; solve(x^2+2*x-15>0,x,'ReturnConditions',true)
Problem 26

Solve: $\displaystyle\frac{x-2}{x+5}\leq 0$

Solution

Critical: $x=2$ (zero OK), $x=-5$ (excluded). Test intervals.

$$\boxed{(-5,2]}$$
syms x; solve((x-2)/(x+5)<=0,x,'ReturnConditions',true)
Problem 27

Solve: $|5x+3|=18$

Solution

$5x+3=18\Rightarrow x=3$. $5x+3=-18\Rightarrow x=-21/5$.

$$\boxed{x=3\text{ or }x=-\tfrac{21}{5}}$$
syms x; solve(abs(5*x+3)==18,x)
Problem 28

Solve: $|3x-2|<7$

Solution

$-7<3x-2<7$, so $-5<3x<9$.

$$\boxed{-\tfrac{5}{3}
syms x; solve(abs(3*x-2)<7,x,'ReturnConditions',true)
Problem 29

Solve: $|2x+1|\geq 9$

Solution

$2x+1\leq -9$ or $2x+1\geq 9$.

$$\boxed{(-\infty,-5]\cup[4,\infty)}$$
syms x; solve(abs(2*x+1)>=9,x,'ReturnConditions',true)
Coordinate Geometry & Lines
Problem 30

Line through $(1,5)$ and $(4,-1)$.

Solution

$m=(-1-5)/(4-1)=-2$. Point-slope: $y-5=-2(x-1)$.

$$\boxed{y=-2x+7}$$
m=(-1-5)/(4-1); b=5-m*1; fprintf('y=%dx+%d\n',m,b)
Problem 31

Line perpendicular to $y=\frac{2}{3}x+4$ through $(6,-1)$.

Solution

$m_\perp=-3/2$. $y+1=-\frac{3}{2}(x-6)$.

$$\boxed{y=-\tfrac{3}{2}x+8}$$
syms x; expand(-3/2*(x-6)+(-1))
Problem 32

Center and radius: $x^2+y^2+10x-4y+13=0$

Solution

$(x^2+10x+25)+(y^2-4y+4)+13-25-4=0$, so $(x+5)^2+(y-2)^2=16$.

$$\boxed{\text{Center }(-5,2),\;\text{radius }4}$$
theta=linspace(0,2*pi,100); plot(-5+4*cos(theta),2+4*sin(theta)); axis equal
Functions & Inverses
Problem 33

$f(x)=x^2+3$, $g(x)=4x-1$. Find $(f\circ g)(x)$.

Solution

$f(g(x))=(4x-1)^2+3=16x^2-8x+1+3$.

$$\boxed{16x^2-8x+4}$$
syms x; f(x)=x^2+3; g(x)=4*x-1; expand(f(g(x)))
Problem 34

Domain of $f(x)=\sqrt{5-3x}$.

Solution

$5-3x\geq 0$, so $x\leq 5/3$.

$$\boxed{\left(-\infty,\tfrac{5}{3}\right]}$$
syms x; solve(5-3*x>=0,x,'ReturnConditions',true)
Problem 35

Inverse of $f(x)=\frac{4x-3}{x+5}$.

Solution

Swap, solve: $x(y+5)=4y-3$, $xy-4y=-5x-3$, $y(x-4)=-(5x+3)$.

$$\boxed{f^{-1}(x)=\frac{-5x-3}{x-4}}$$
syms x; finverse((4*x-3)/(x+5))
Problem 36

Even, odd, or neither: $f(x)=x^3-5x$

Solution

$f(-x)=-x^3+5x=-(x^3-5x)=-f(x)$.

$$\boxed{\text{Odd (origin symmetry)}}$$
syms x; f=x^3-5*x; isequal(simplify(subs(f,x,-x)),simplify(-f))
Problem 37

Describe transformations: $g(x)=-2(x-4)^2+7$ relative to $f(x)=x^2$.

Solution

Shift right 4, vertical stretch ×2, reflect over $x$-axis, shift up 7. Vertex $(4,7)$, opens down.

Conic Sections & Rational Functions
Problem 38

Vertex form: $f(x)=3x^2+18x+23$

Solution

$3(x^2+6x)+23=3(x+3)^2-27+23$.

$$\boxed{3(x+3)^2-4}\text{, vertex }(-3,-4)$$
syms x; c=sym2poly(3*x^2+18*x+23); xv=-c(2)/(2*c(1))
Problem 39

Center, semi-axes, foci: $\frac{(x-2)^2}{25}+\frac{(y+1)^2}{16}=1$

Solution

Center $(2,-1)$, $a=5$, $b=4$, $c=\sqrt{25-16}=3$. Horizontal major axis.

$$\boxed{\text{Foci: }(-1,-1)\text{ and }(5,-1)}$$
a=5;b=4;c=sqrt(a^2-b^2) % 3
Problem 40

Asymptotes and foci: $\frac{x^2}{4}-\frac{y^2}{25}=1$

Solution

$a=2$, $b=5$. Asymptotes: $y=\pm\frac{5}{2}x$. $c=\sqrt{4+25}=\sqrt{29}$.

$$\boxed{\text{Foci: }(\pm\sqrt{29},0)}$$
a=2;b=5;c=sqrt(a^2+b^2) % sqrt(29)
Problem 41

All asymptotes: $f(x)=\frac{3x^2-x+2}{x^2-1}$

Solution

VA: $x=\pm 1$. HA: degrees equal, $y=3/1=3$.

$$\boxed{\text{VA: }x=\pm 1,\;\text{HA: }y=3}$$
syms x; limit((3*x^2-x+2)/(x^2-1),x,inf) % 3
Problem 42

Slant asymptote: $g(x)=\frac{x^2-2x+3}{x+1}$

Solution

Long division: $\frac{x^2-2x+3}{x+1}=x-3+\frac{6}{x+1}$. VA: $x=-1$.

$$\boxed{\text{Slant: }y=x-3,\;\text{VA: }x=-1}$$
syms x; [q,r]=quorem(x^2-2*x+3,x+1,x) % q=x-3
Polynomial Analysis & Partial Fractions
Problem 43

Divide: $(3x^3+2x^2-5x+4)\div(x+2)$

Solution
$$\boxed{3x^2-4x+3\text{, remainder }-2}$$
syms x; [q,r]=quorem(3*x^3+2*x^2-5*x+4,x+2,x)
Problem 44

$x=2$ is a root of $f(x)=x^3-x^2-8x+12$. Factor completely.

Solution

Divide by $(x-2)$: $x^2+x-6=(x+3)(x-2)$.

$$\boxed{(x-2)^2(x+3)}$$
syms x; factor(x^3-x^2-8*x+12)
Problem 45

Find all rational roots: $2x^3+3x^2-8x+3=0$

Solution

Test $x=1$: $2+3-8+3=0$ ✓. Factor: $(x-1)(2x-1)(x+3)$.

$$\boxed{x=1,\;\frac{1}{2},\;-3}$$
syms x; solve(2*x^3+3*x^2-8*x+3==0,x)
Problem 46

Decompose: $\frac{7x-1}{(x-2)(x+3)}$

Solution

$x=2$: $13=5A$, $A=13/5$. $x=-3$: $-22=-5B$, $B=22/5$.

$$\boxed{\frac{13/5}{x-2}+\frac{22/5}{x+3}}$$
syms x; partfrac((7*x-1)/((x-2)*(x+3)))
Problem 47

Decompose: $\frac{2x^2+x-1}{(x+1)(x^2+4)}$

Solution

$A/(x+1)+(Bx+C)/(x^2+4)$. At $x=-1$: $A=0$. Expand and compare: $B=2$, $C=-1$.

$$\boxed{\frac{2x-1}{x^2+4}}$$
syms x; partfrac((2*x^2+x-1)/((x+1)*(x^2+4)))
Exponentials & Logarithms
Problem 48

Evaluate: $\log_4 64$, $\log_{1/3}9$, $\ln e^7$

Solution

$4^3=64\Rightarrow 3$. $(1/3)^{-2}=9\Rightarrow -2$. $\ln e^7=7$.

$$\boxed{3,\;-2,\;7}$$
log(64)/log(4) % 3; log(9)/log(1/3) % -2; log(exp(7)) % 7
Problem 49

Expand: $\ln\left(\frac{x^4\sqrt{y}}{z^3}\right)$

Solution
$$\boxed{4\ln x+\tfrac{1}{2}\ln y-3\ln z}$$

LaTeX: 4\ln x + \frac{1}{2}\ln y - 3\ln z

syms x y z; expand(log(x^4*sqrt(y)/z^3))
Problem 50

Solve: $4^{x+1}=32$

Solution

$2^{2(x+1)}=2^5$, so $2x+2=5$.

$$\boxed{x=\frac{3}{2}}$$
syms x; solve(4^(x+1)==32,x)
Problem 51

Solve: $e^{2x}-7e^x+10=0$

Solution

Let $u=e^x$: $(u-2)(u-5)=0$.

$$\boxed{x=\ln 2\text{ or }x=\ln 5}$$
syms x; solve(exp(2*x)-7*exp(x)+10==0,x)
Problem 52

Solve: $\log_3(x+1)+\log_3(x-3)=3$

Solution

$(x+1)(x-3)=27$, $x^2-2x-30=0$, $x=1\pm\sqrt{31}$. Domain: $x>3$, only $1+\sqrt{31}\approx 6.57$ ✓.

$$\boxed{x=1+\sqrt{31}}$$
syms x; solve(log(x+1)/log(3)+log(x-3)/log(3)==3,x)
Problem 53

$8000 at 5% compounded quarterly. Time to reach $15000?

Solution

$15000=8000(1.0125)^{4t}$, $\ln 1.875=4t\ln 1.0125$.

$$\boxed{t=\frac{\ln 1.875}{4\ln 1.0125}\approx 12.63\text{ years}}$$
t=log(15000/8000)/(4*log(1.0125)) % 12.63
Systems of Equations
Problem 54

Solve: $5x+2y=11$, $3x-4y=1$

Solution

Multiply first by 2, add: $13x=23$.

$$\boxed{x=\frac{23}{13},\;y=\frac{14}{13}}$$
A=[5 2;3 -4]; b=[11;1]; x=A\b
Problem 55

RREF: $x+3y-z=4$, $2x-y+2z=1$, $3x+2y+z=5$

Solution
$$\left[\begin{array}{ccc|c}1&3&-1&4\\2&-1&2&1\\3&2&1&5\end{array}\right]\xrightarrow{\text{RREF}}\left[\begin{array}{ccc|c}1&0&0&1\\0&1&0&1\\0&0&1&0\end{array}\right]$$
$$\boxed{x=1,\;y=1,\;z=0}$$
rref([1 3 -1 4;2 -1 2 1;3 2 1 5])
Problem 56

Solve: $x^2+y^2=20$, $y=x+2$

Solution

$x^2+(x+2)^2=20$, $2x^2+4x-16=0$, $(x+4)(x-2)=0$.

$$\boxed{(-4,-2)\text{ and }(2,4)}$$
syms x y; solve([x^2+y^2==20,y==x+2],[x,y])
Problem 57

Solve: $xy=18$, $x+y=9$

Solution

$x(9-x)=18$, $x^2-9x+18=0$, $(x-3)(x-6)=0$.

$$\boxed{(3,6)\text{ and }(6,3)}$$
syms x y; solve([x*y==18,x+y==9],[x,y])
Problem 58

A boat: 48 km downstream in 3 hrs, 48 km upstream in 4 hrs. Find boat speed and current.

Solution

$b+c=16$, $b-c=12$. Add: $2b=28$.

$$\boxed{b=14\text{ km/h},\;c=2\text{ km/h}}$$
syms b c; solve([b+c==16,b-c==12],[b,c])
Problem 59

Solve: $\frac{x^2}{4}+\frac{y^2}{9}=1$, $y=x+1$

Solution

$9x^2+4(x+1)^2=36$, $13x^2+8x-32=0$.

$$\boxed{x=\frac{-4\pm 12\sqrt{3}}{13}}$$
syms x y; solve([x^2/4+y^2/9==1,y==x+1],[x,y])
Problem 60

Half-life 6 hours, starting 500g. How much after 15 hours?

Solution

$N=500\cdot 2^{-15/6}=500\cdot 2^{-2.5}=500/(4\sqrt{2})$.

$$\boxed{\approx 88.39\text{ g}}$$

LaTeX: N(t)=N_0 e^{-t\ln 2/t_{1/2}}

N0=500;t_half=6;t=15; N=N0*exp(-log(2)*t/t_half) % 88.39
Algebra Practice Problems — 60 Original Problems
Solutions with LaTeX syntax and MATLAB verification