# 2.6 Other types of equations  (Page 2/10)

 Page 2 / 10

Solve: $\text{\hspace{0.17em}}{\left(x+5\right)}^{\frac{3}{2}}=8.$

$\left\{-1\right\}$

## Solving equations using factoring

We have used factoring to solve quadratic equations, but it is a technique that we can use with many types of polynomial equations, which are equations that contain a string of terms including numerical coefficients and variables. When we are faced with an equation containing polynomials of degree higher than 2, we can often solve them by factoring.

## Polynomial equations

A polynomial of degree n is an expression of the type

${a}_{n}{x}^{n}+{a}_{n-1}{x}^{n-1}+\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}$

where n is a positive integer and $\text{\hspace{0.17em}}{a}_{n},\dots ,{a}_{0}\text{\hspace{0.17em}}$ are real numbers and $\text{\hspace{0.17em}}{a}_{n}\ne 0.$

Setting the polynomial equal to zero gives a polynomial equation    . The total number of solutions (real and complex) to a polynomial equation is equal to the highest exponent n .

## Solving a polynomial by factoring

Solve the polynomial by factoring: $\text{\hspace{0.17em}}5{x}^{4}=80{x}^{2}.$

First, set the equation equal to zero. Then factor out what is common to both terms, the GCF.

$\begin{array}{ccc}\hfill 5{x}^{4}-80{x}^{2}& =& 0\hfill \\ \hfill 5{x}^{2}\left({x}^{2}-16\right)& =& 0\hfill \end{array}$

Notice that we have the difference of squares in the factor $\text{\hspace{0.17em}}{x}^{2}-16,$ which we will continue to factor and obtain two solutions. The first term, $\text{\hspace{0.17em}}5{x}^{2},$ generates, technically, two solutions as the exponent is 2, but they are the same solution.

$\begin{array}{ccc}\hfill 5{x}^{2}& =& 0\hfill \\ \hfill x& =& 0\hfill \\ \hfill {x}^{2}-16& =& 0\hfill \\ \hfill \left(x-4\right)\left(x+4\right)& =& 0\hfill \\ \hfill x& =& 4\hfill \\ \hfill x& =& -4\hfill \end{array}$

The solutions are $4,$ and $\text{\hspace{0.17em}}-4.$

Solve by factoring: $\text{\hspace{0.17em}}12{x}^{4}=3{x}^{2}.$

$x=0,$ $x=\frac{1}{2},$ $x=-\frac{1}{2}$

## Solve a polynomial by grouping

Solve a polynomial by grouping: $\text{\hspace{0.17em}}{x}^{3}+{x}^{2}-9x-9=0.$

This polynomial consists of 4 terms, which we can solve by grouping. Grouping procedures require factoring the first two terms and then factoring the last two terms. If the factors in the parentheses are identical, we can continue the process and solve, unless more factoring is suggested.

$\begin{array}{ccc}\hfill {x}^{3}+{x}^{2}-9x-9& =& 0\hfill \\ \hfill {x}^{2}\left(x+1\right)-9\left(x+1\right)& =& 0\hfill \\ \hfill \left({x}^{2}-9\right)\left(x+1\right)& =& 0\hfill \end{array}$

The grouping process ends here, as we can factor $\text{\hspace{0.17em}}{x}^{2}-9\text{\hspace{0.17em}}$ using the difference of squares formula.

$\begin{array}{ccc}\left({x}^{2}-9\right)\left(x+1\right)& =& 0\hfill \\ \hfill \left(x-3\right)\left(x+3\right)\left(x+1\right)& =& 0\hfill \\ \hfill x& =& 3\hfill \\ \hfill x& =& -3\hfill \\ \hfill x& =& -1\hfill \end{array}$

The solutions are $3,$ $-3,$ and $\text{\hspace{0.17em}}-1.\text{\hspace{0.17em}}$ Note that the highest exponent is 3 and we obtained 3 solutions. We can see the solutions, the x- intercepts, on the graph in [link] .

## Solving radical equations

Radical equations are equations that contain variables in the radicand    (the expression under a radical symbol), such as

$\begin{array}{ccc}\hfill \sqrt{3x+18}& =& x\hfill \\ \hfill \sqrt{x+3}& =& x-3\hfill \\ \hfill \sqrt{x+5}-\sqrt{x-3}& =& 2\hfill \end{array}$

Radical equations may have one or more radical terms, and are solved by eliminating each radical, one at a time. We have to be careful when solving radical equations, as it is not unusual to find extraneous solutions    , roots that are not, in fact, solutions to the equation. These solutions are not due to a mistake in the solving method, but result from the process of raising both sides of an equation to a power. However, checking each answer in the original equation will confirm the true solutions.

An equation containing terms with a variable in the radicand is called a radical equation    .

Given a radical equation, solve it.

1. Isolate the radical expression on one side of the equal sign. Put all remaining terms on the other side.
2. If the radical is a square root, then square both sides of the equation. If it is a cube root, then raise both sides of the equation to the third power. In other words, for an n th root radical, raise both sides to the n th power. Doing so eliminates the radical symbol.
3. Solve the remaining equation.
4. If a radical term still remains, repeat steps 1–2.
5. Confirm solutions by substituting them into the original equation.

#### Questions & Answers

find general solution of the Tanx=-1/root3,secx=2/root3
find general solution of the following equation
Nani
the value of 2 sin square 60 Cos 60
0.75
Lynne
0.75
Inkoom
when can I use sin, cos tan in a giving question
depending on the question
Nicholas
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
John
I want to learn the calculations
where can I get indices
I need matrices
Nasasira
hi
Raihany
Hi
Solomon
need help
Raihany
maybe provide us videos
Nasasira
Raihany
Hello
Cromwell
a
Amie
What do you mean by a
Cromwell
nothing. I accidentally press it
Amie
you guys know any app with matrices?
Khay
Ok
Cromwell
Solve the x? x=18+(24-3)=72
x-39=72 x=111
Suraj
Solve the formula for the indicated variable P=b+4a+2c, for b
Need help with this question please
b=-4ac-2c+P
Denisse
b=p-4a-2c
Suddhen
b= p - 4a - 2c
Snr
p=2(2a+C)+b
Suraj
b=p-2(2a+c)
Tapiwa
P=4a+b+2C
COLEMAN
b=P-4a-2c
COLEMAN
like Deadra, show me the step by step order of operation to alive for b
John
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
The sequence is {1,-1,1-1.....} has
circular region of radious
how can we solve this problem
Sin(A+B) = sinBcosA+cosBsinA
Prove it
Eseka
Eseka
hi
Joel
yah
immy
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
7.5 and 37.5
Nando
how would this look as an equation?
Hayden
5x+x=45
Khay
find the sum of 28th term of the AP 3+10+17+---------
I think you should say "28 terms" instead of "28th term"
Vedant
the 28th term is 175
Nando
192
Kenneth
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n By By   By  By By    By