Algebra 3 Quiz: Solving Rational Equations

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Solving Rational EquationsQuestion 1 of 20

Solve the rational equation: $\frac{x}{x+2}+\frac{2}{x-2}=\frac{x^2+4}{x^2-4}.$

The only solution is

x=0

All real numbers except

-2

and

2

There is no solution.

The only solution is

x=2

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Algebra 3 Quiz

Algebra 3 Quiz: Solving Rational Equations

Practice Solving Rational Equations in Algebra 3 with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Solving Rational Equations, giving you a quick way to practice the rules, question types, and explanations that matter most for Algebra 3.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Solve the rational equation: $\frac{x}{x+2}+\frac{2}{x-2}=\frac{x^2+4}{x^2-4}.$

The only solution is $x=0$ .
All real numbers except $-2$ and $2$ . (correct answer)
There is no solution.
The only solution is $x=2$ .

Explanation: Whenever you solve a rational equation, your first move should always be to identify the domain: any value that makes a denominator zero cannot be a solution. Here the denominators are

x+2

x-2

, and

x^2-4=(x+2)(x-2)

, so

x=-2

and

x=2

are immediately excluded. Now combine the left side:

\frac{x}{x+2}+\frac{2}{x-2}=\frac{x(x-2)+2(x+2)}{x^2-4}=\frac{x^2-2x+2x+4}{x^2-4}=\frac{x^2+4}{x^2-4}.

That is exactly the right side. So the equation is an identity for every allowable value of

x

. Since the only excluded values are

-2

and

2

, the solution set is all real numbers except those two. Why are the other choices traps? "The only solution is

x=0

" is tempting because

0

does satisfy the equation, but it is not the only solution — any allowable number works. "There is no solution" might come from noticing the excluded values, but that confuses "not defined" with "no solution." "The only solution is

x=2

" is doubly wrong:

x=2

makes the denominator

x-2

and

x^2-4

zero, so it is not even in the domain. Study tip: before doing any algebra in a rational equation, write down the domain restrictions. Then simplify both sides. If they become identical, the solution is every domain value — and if your simplification creates an answer outside the domain, discard it as extraneous.

Question 2

A student solves $\frac{x}{x-1}+\frac{2}{x+1}=\frac{7x-5}{x^2-1}$ as follows: Step 1: $x(x+1)+2(x-1)=7x-5$ . Step 2: $x^2+3x-2=7x-5$ . Step 3: $x^2-4x+3=0$ . Step 4: $(x-1)(x-3)=0$ , so $x=1$ or $x=3$ . Step 5: Since both values satisfy the multiplied equation, the solutions are $x=1$ and $x=3$ . At which step is the first error?

Step 1
Step 3
Step 4
Step 5 (correct answer)

Explanation: Whenever you solve a rational equation, multiplying by the LCD clears denominators, but it can also introduce extraneous solutions that make an original denominator zero. Here the LCD is

(x-1)(x+1)

, so the domain requires

x\neq 1

and

x\neq -1

. Step 1 correctly multiplies each term by the LCD to get

x(x+1)+2(x-1)=7x-5

. Step 2 expands correctly to

x^2+3x-2=7x-5

, and Step 3 correctly rearranges to

x^2-4x+3=0

. Step 4 factors correctly as

(x-1)(x-3)=0

, giving

x=1

x=3

. All these algebraic steps are valid. The first error is in Step 5, where the student says both values are solutions because they satisfy the multiplied equation. But

x=1

makes the original denominators

x-1

and

x^2-1

equal to zero, so it is extraneous and must be rejected. The only valid solution is

x=3

. The choice "Step 1" is a trap if you think the multiplication is wrong; it is correct, but you still must check domain restrictions afterward. "Step 3" and "Step 4" are not errors because the rearrangement and factoring are accurate. Study tip: after solving a rational equation, always plug each candidate into the original denominators; any value that gives zero is extraneous, no matter how nicely it satisfies the cleared equation.

Question 3

Which equation has $x=2$ as an extraneous solution?

$\frac{x^2-4}{x+2}=0$
$\frac{x}{x-2}+2=\frac{2}{x-2}$ (correct answer)
$\frac{x^2-3x+2}{x-1}=0$
$\frac{x^2-5x+6}{x-3}=0$

Explanation: Whenever you solve a rational equation, extraneous solutions are numbers that appear from the algebra but make a denominator zero in the original equation. Your first move should be to note which

x

-values are excluded. For

\frac{x}{x-2}+2=\frac{2}{x-2}

, the denominator

x-2

excludes

x=2

. If you multiply both sides by

x-2

, you get

x+2(x-2)=2

, which simplifies to

3x=6

, so

x=2

. But

x=2

was excluded, so it is extraneous. Now check the others. In

\frac{x^2-4}{x+2}=0

, factoring gives

\frac{(x-2)(x+2)}{x+2}=0

, and after canceling,

x=2

. Since

x+2\neq 0

x=2

, it is a valid solution. Similarly,

\frac{x^2-3x+2}{x-1}=0

factors to

\frac{(x-1)(x-2)}{x-1}=0

, giving

x=2

, and

x-1\neq0

x=2

. And

\frac{x^2-5x+6}{x-3}=0

factors to

\frac{(x-2)(x-3)}{x-3}=0

, again giving

x=2

, with

x-3\neq0

x=2

. Each of these has a different excluded value, but none excludes

2

. Study tip: before solving a rational equation, write down excluded values. After solving, reject any candidate that is excluded. Extraneous solutions come from multiplying by a variable denominator, not from the factoring itself.

Question 4

A rational equation has domain restrictions $x\neq 1$ and $x\neq -2$ . After solving, a student obtains $x=1$ , $x=-2$ , and $x=3$ . Which statement is true?

The solution set is $\{1,-2\}$ .
The solution set is $\{1,-2,3\}$ .
The solution set is $\{3\}$ . (correct answer)
The equation has no solution.

Explanation: When you solve a rational equation, your first step is always to note the domain restrictions: any value that makes a denominator zero cannot be a solution. Here, the restrictions are

x\neq 1

and

x\neq -2

. Once you solve and get candidate answers, you must filter them through those restrictions. The student found three candidates:

x=1

x=-2

, and

x=3

. But

x=1

and

x=-2

are exactly the values that are excluded from the domain, so they cannot be solutions. That leaves

x=3

, which is allowed, so the solution set is indeed

\{3\}

. Now look at the wrong choices. The set

\{1,-2\}

contains only the two forbidden values, so it is not a valid solution set. The set

\{1,-2,3\}

incorrectly includes the restricted values along with the valid one. And the claim that the equation has no solution is false because

x=3

satisfies the equation and is permitted by the domain. The trap here is forgetting to check your solutions against the original domain restrictions — especially when the algebra itself produces those restricted values. Your study takeaway: after solving any rational equation, always substitute your answers back into the original denominators or compare them to the stated domain. A value that makes a denominator zero is never a solution, even if it appears in your factored work. That simple check is often the difference between a correct solution set and a careless mistake.

Question 5

For which value of $k$ does $\frac{x}{x-3}+\frac{k}{x+3}=\frac{18}{x^2-9}$ have exactly one distinct algebraic solution, and that solution is extraneous?

$-9$ (correct answer)
$-3$
$3$
$9$

Explanation: When you solve a rational equation like this, the first move is always: identify the excluded values, then multiply by the LCD

(x-3)(x+3)

to clear denominators. Here

x=3

and

x=-3

are forbidden. After multiplying, you get

x^2+(k+3)x-3k-18=0.

For the equation to have exactly one distinct algebraic solution, this quadratic must have a double root. Its discriminant is

(k+3)^2-4(-3k-18)=(k+9)^2,

so the only way to get one distinct solution is

k=-9

. Substituting

k=-9

gives

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

x=3

. But

x=3

makes the original denominator

x-3

zero, so the single solution is extraneous. Thus

k=-9

satisfies both conditions. For

k=-3

, the cleared equation is

x^2-9=0

, giving

x=\pm3

, both extraneous — two algebraic solutions, not one. For

k=3

, you get

(x+9)(x-3)=0

, so

x=3

is extraneous but

x=-9

is valid. For

k=9

, you get

(x-3)(x+15)=0

, so again

x=3

is extraneous but

x=-15

is valid. Each of those choices leaves a valid solution, so they miss the "extraneous-only" requirement. When you see "exactly one distinct algebraic solution and it is extraneous," think: the cleared equation must have a repeated root, and that repeated root must be a forbidden value. Check the discriminant and the domain.

Question 6

After correctly multiplying $\frac{x}{x+2}+\frac{3}{x-2}=\frac{3x+14}{x^2-4}$ by the LCD, a student obtains $x^2-2x-8=0$ and concludes that $x=4$ or $x=-2$ . What is the solution set?

$\{-2\}$
$\{-2,4\}$
$\{4\}$ (correct answer)
$\varnothing$

Explanation: Whenever you solve a rational equation by multiplying both sides by the LCD, you're actually assuming the denominators are not zero. That's why this question tests extraneous solutions: after solving the polynomial, you must check every candidate back in the original denominators. Here, the LCD is

(x+2)(x-2)=x^2-4

. Multiplying correctly clears the fractions and gives

x^2-2x-8=0

. Factoring yields

(x-4)(x+2)=0

, so the candidates are

x=4

and

x=-2

. But

x=-2

would make the denominator

x+2

equal zero—and also make

x^2-4

zero on the right-hand side—so it is undefined in the original equation. Therefore

x=-2

is extraneous and must be discarded. On the other hand,

x=4

gives denominators

6

and

2

, both nonzero, and it satisfies the equation:

\frac46+\frac32=\frac{13}{6}

, and the right side is

\frac{26}{12}=\frac{13}{6}

. So the solution set is

\{4\}

. The choice

\{-2\}

misses the valid root; the choice

\{-2,4\}

includes the extraneous root because it stopped at the polynomial equation instead of checking domain restrictions; and the choice

\varnothing

would mean no valid solution, but

x=4

definitely works. Study tip: on any rational equation, after solving the cleared equation, always test each candidate in the original denominators—any root that makes a denominator zero is extraneous, no matter how nicely it solved the polynomial.

Question 7

Solve the rational equation: $\frac{x}{x-2}+\frac{3}{x-4}=\frac{5x-14}{(x-2)(x-4)}.$

$x=2$ and $x=4$
The equation has no solution (correct answer)
The only solution is $x=2$
The only solution is $x=4$

Explanation: When you see a rational equation, your first move should always be to identify the domain restrictions—values that make any denominator zero. Here,

x=2

and

x=4

are off-limits because they would cause division by zero. To solve, multiply both sides by the least common denominator

(x-2)(x-4)

. This gives

x(x-4)+3(x-2)=5x-14

, which simplifies to

x^2-6x+8=0

. Factoring yields

(x-2)(x-4)=0

, so

x=2

x=4

. But both of these are exactly the values excluded from the domain. Substituting them into the original equation would make the denominators zero, so neither is a valid solution. That means the equation has no solution. The trap choices reflect common mistakes. Choosing "

x=2

and

x=4

" comes from solving the factored equation without checking the domain—these are extraneous solutions. "The only solution is

x=2

" or "The only solution is

x=4

" each pick one of the extraneous roots, likely from a sign error or forgetting to check the other root. All three fail to recognize that the domain restrictions invalidate every algebraic solution. Your takeaway: for any rational equation, list excluded values first. After solving, always verify that each candidate does not violate those restrictions. If every solution you find is excluded, the answer is "no solution"—not a valid root. This step is quick but prevents the most common error on this type of exam question.

Question 8

What is the solution set of $\frac{x+1}{x-1}-\frac{2}{x+2}=\frac{6}{x^2+x-2}$ ?

$\{-2\}$
$\{-2,1\}$
$\varnothing$ (correct answer)
$\{1\}$

Explanation: Whenever you solve a rational equation, start by finding domain restrictions: denominators cannot be zero. Here

x-1

and

x+2

appear, so

x\neq 1

and

x\neq -2

. Since

x^2+x-2=(x-1)(x+2)

, the right side gives the same restrictions. Multiply through by the common denominator

(x-1)(x+2)

. The left side becomes

(x+1)(x+2)-2(x-1)=x^2+x+4

. Setting this equal to

6

gives

x^2+x+4=6

, so

x^2+x-2=0

, which factors as

(x+2)(x-1)=0

. The algebra gives

x=-2

x=1

. But both of these values are excluded from the domain, so neither can be a solution. Therefore the solution set is the empty set

\varnothing

. The choices

\{-2\}

and

\{1\}

each contain one excluded value, and

\{-2,1\}

contains both, but all three ignore the fact that these values make original denominators zero. The empty set is the only valid answer. Study tip: on rational equations, write the excluded values before solving, then discard any solution that matches one of them. Checking the domain is not optional—it is exactly what separates the correct answer from these traps.

Question 9

Solve the rational equation: $\frac{x}{x-3}+\frac{2}{x+3}=\frac{18}{x^2-9}.$

$x=-8$ only (correct answer)
$x=-8$ and $x=3$
$x=3$ only
$x=8$ only

Explanation: When you solve a rational equation, your first step should be to factor denominators and note excluded values. Here,

x^2-9=(x-3)(x+3)

, so

x

cannot be

3

-3

. Multiply every term by the least common denominator

(x-3)(x+3)

x(x+3)+2(x-3)=18.

Expanding gives

x^2+3x+2x-6=18

, so

x^2+5x-24=0

. Factor:

(x+8)(x-3)=0

, giving potential solutions

x=-8

and

x=3

. But

x=3

makes the original denominators

x-3

and

x^2-9

equal to zero, so it is extraneous. Therefore, the valid solution is $x=-8$ only. The choice "

x=-8

and

x=3

" is the trap: it includes the extraneous root because the solving process was not checked against the original denominators. "

x=3

only" is also wrong—it keeps the extraneous value and discards the valid

-8

. "

x=8

only" comes from a sign error; substituting

x=8

gives

\frac{8}{5}+\frac{2}{11}=\frac{98}{55}

, not

\frac{18}{55}

. Your strategy for rational equations: find excluded values before clearing fractions, solve the resulting polynomial, then always check each candidate against the original equation. On the exam, extraneous roots are the most common trap—if a solution makes any denominator zero, it must be discarded.

Question 10

Solve the rational equation: $\frac{x^2-4}{x-2}=0.$

There is no solution.
$x=2$ and $x=-2$
$x=2$
$x=-2$ (correct answer)

Explanation: When solving a rational equation, always keep the denominator in mind: a value that makes the denominator zero is not allowed, even if it makes the numerator zero. Here you have

\frac{x^2-4}{x-2}=0.

Factor the numerator as a difference of squares:

x^2-4=(x-2)(x+2).

So the equation becomes

\frac{(x-2)(x+2)}{x-2}=0.

You might be tempted to cancel

x-2

, but cancellation is only valid when

x\neq 2

, since plugging

x=2

into the original equation gives a denominator of zero. Assuming

x\neq 2

, the expression simplifies to

x+2=0,

x=-2.

Since

-2

does not make the original denominator zero, it is the valid solution. The choice "

x=2

and

x=-2

" captures the trap of solving the numerator alone:

x=2

makes the numerator zero, but it also makes the denominator zero, so the original expression is undefined there. The choice "

x=2

" alone has the same flaw. The choice "there is no solution" is incorrect because

-2

satisfies the original equation:

\frac{(-2)^2-4}{-2-2}=\frac{0}{-4}=0.

On rational equation problems, write down excluded values from the denominator before simplifying. Then solve the simplified equation and check that your answer is not one of those excluded values. That single habit prevents the most common errors on this exam.

Question 11

What is the solution set of $\frac{x}{x-2}+\frac{3}{x+4}=\frac{6x}{x^2+2x-8}$ ?

$\{-3,2\}$
$\{-3\}$ (correct answer)
$\{2\}$
$\varnothing$

Explanation: When you solve a rational equation, the key is to remember that multiplying by a common denominator can introduce extraneous solutions — values that satisfy the transformed equation but make an original denominator zero. So first note the domain:

x\neq 2

and

x\neq -4

, because either would make a denominator undefined. The denominator

x^2+2x-8

factors as

(x-2)(x+4)

, so the least common denominator is

(x-2)(x+4)

. Multiply every term by this:

x(x+4)+3(x-2)=6x

x^2+4x+3x-6=6x

x^2+7x-6=6x

x^2+x-6=0

(x+3)(x-2)=0

So the potential solutions are

x=-3

and

x=2

. But

x=2

is excluded from the domain because it makes

x-2

zero, so it is extraneous. The only valid solution is

-3

, giving

\{-3\}

. The choice

\{-3,2\}

includes the extraneous value

2

, so it fails the domain check. The choice

\{2\}

is entirely extraneous — it comes from solving but violates the original equation. The choice

\varnothing

misses that

-3

is perfectly valid after substitution. A good habit: before solving any rational equation, write down the excluded values. Then after solving, always check your answers against that list. This simple step prevents the most common trap on this exam.

Question 12

A boat travels 10 miles upstream and 10 miles downstream in a total of 5 hours. The current is 2 miles per hour. If $b$ is the boat's speed in still water, the situation is modeled by $\frac{10}{b-2}+\frac{10}{b+2}=5$ . What is $b$ ?

$2+2\sqrt{2}$ miles per hour (correct answer)
$2-2\sqrt{2}$ miles per hour
$4$ miles per hour
$2+\sqrt{2}$ miles per hour

Explanation: Whenever you see a distance-rate-time word problem modeled by a rational equation, your two-step plan should be: solve the equation, then check every solution against the physical situation. Here the boat must make positive progress upstream, so

b>2

; any root less than 2 is invalid. Multiplying

\frac{10}{b-2}+\frac{10}{b+2}=5

(b-2)(b+2)

gives

10(b+2)+10(b-2)=5(b^2-4)

. Simplify to

20b=5b^2-20

, so

b^2-4b-4=0

. The quadratic formula yields

b=\frac{4\pm\sqrt{32}}{2}=2\pm2\sqrt{2}

. The positive value

2+2\sqrt2

satisfies

b>2

and is the correct speed. The other algebraic root,

2-2\sqrt2

, is a negative number, so it cannot be a boat speed; it also violates

b>2

. Choosing

4

mph might come from assuming the current cancels completely, but substituting gives

10/2+10/6=6\frac23

hours, not 5. Choosing

2+\sqrt2

is an arithmetic slip:

\sqrt{32}/2

2\sqrt2

, not

\sqrt2

, so this underestimates the correction to the still-water speed. On exam day, after solving a rational equation, always plug your answer back into the original word-problem conditions. Time must be positive, upstream speed must exceed the current, and any algebraic extraneous or negative root should be discarded.

Question 13

For what value of $a$ does $\frac{2x+3}{x-1}-\frac{x}{x+2}=\frac{a}{x^2+x-2}$ have no real solution?

$-12$ (correct answer)
$-10$
$-8$
$8$

Explanation: Since

x^2+x-2=(x-1)(x+2)

, the domain excludes

x=1

and

x=-2

. Multiplying by the LCD gives

(2x+3)(x+2)-x(x-1)=a

, or

x^2+8x+6=a

, so

x^2+8x+(6-a)=0

. Its discriminant is

64-4(6-a)=40+4a

. For no real solution,

40+4a<0

, so

a<-10

. Only

-12

satisfies this among the choices.

Question 14

Which statement correctly describes the real solutions of $\frac{x}{x+1}+\frac{x+1}{x}=\frac{2x^2+1}{x^2+x}$ ?

The only real solution is $x=-1$ .
The only real solution is $x=0$ .
The equation has no real solution. (correct answer)
The equation has two real solutions, $x=0$ and $x=-1$ .

Explanation: The domain excludes

x=0

and

x=-1

. Over the common denominator

x(x+1)

, the left side becomes

\frac{2x^2+2x+1}{x^2+x}

. Setting numerators equal gives

2x^2+2x+1=2x^2+1

, so

x=0

. Since

x=0

is not in the domain, the original equation has no real solution.

Question 15

How many real numbers satisfy $\frac{x^2-1}{x-1}=\frac{x^2+x}{x}$ ?

None
Exactly one
Exactly two
Infinitely many (correct answer)

Explanation: The domain excludes

x=1

because of

x-1

, and

x=0

because of the denominator

x

. For every other real value,

\frac{x^2-1}{x-1}=x+1

and

\frac{x^2+x}{x}=x+1

. Thus the equation is an identity for all real numbers except

0

and

1

, so infinitely many real numbers satisfy it.

Question 16

For what value of $k$ is $x=-3$ an extraneous solution of $\frac{x}{x+3}-\frac{5}{x-1}=\frac{k}{x^2+2x-3}$ ?

$-12$
$-6$
$6$
$12$ (correct answer)

Explanation: Factor

x^2+2x-3=(x+3)(x-1)

, so

x=-3

and

x=1

are excluded. Multiplying by the LCD gives

x(x-1)-5(x+3)=k

. Substituting

x=-3

yields

(-3)(-4)-5(0)=12

, so

k=12

. With

k=12

, the cleared equation has roots

9

and

-3

;

-3

is extraneous, leaving

9

valid.

Question 17

What is the solution set of $\frac{1}{x-1}+\frac{2}{x^2+x+1}=\frac{3x}{x^3-1}$ ?

$\varnothing$
$\{-1\}$ (correct answer)
$\{1\}$
$\{-1,1\}$

Explanation: Factor

x^3-1=(x-1)(x^2+x+1)

, and

x^2+x+1

is never zero, so

x=1

is the only excluded value. Multiplying by

x^3-1

gives

x^2+x+1+2(x-1)=3x

, which simplifies to

x^2-1=0

. Thus the candidates are

x=\pm1

; since

x=1

is excluded, the solution set is

\{-1\}

Question 18

What is the solution set of $\frac{x}{x-2}+\frac{x+1}{x}=\frac{4x-2}{x(x-2)}$ ?

$\left\{\frac52\right\}$ (correct answer)
$\{0\}$
$\left\{0,\frac52\right\}$
$\varnothing$

Explanation: The domain excludes

x=0

and

x=2

. Multiplying by

x(x-2)

gives

x^2+(x+1)(x-2)=4x-2

, which simplifies to

2x^2-5x=0

, so

x=0

x=\frac52

. Since

0

is excluded, the only valid solution is

\frac52

Question 19

A bus rental costs ($120). If $n$ students attend, the cost per student is $\frac{120}{n}$ dollars. If 4 additional students attend, the cost per student decreases by ($1). This situation is modeled by $\frac{120}{n}=\frac{120}{n+4}+1$ . How many students are in the original group?

$-24$
$-20$
$20$ (correct answer)
$24$

Explanation: Multiplying by

n(n+4)

gives

120(n+4)=120n+n(n+4)

, so

120n+480=120n+n^2+4n

, or

n^2+4n-480=0

. Factoring gives

(n+24)(n-20)=0

, so

n=-24

n=20

. Since a student count cannot be negative, the original group has

20

students.

Question 20

Which equation has solution set $\{-1\}$ after extraneous solutions are rejected?

$\frac{x}{x-2}-\frac{4}{x+3}=\frac{10}{(x-2)(x+3)}$ (correct answer)
$\frac{x}{x-2}-\frac{4}{x+3}=\frac{12}{(x-2)(x+3)}$
$\frac{x}{x-2}+\frac{4}{x+3}=\frac{10}{(x-2)(x+3)}$
$\frac{x}{x-2}-\frac{2}{x+3}=\frac{10}{(x-2)(x+3)}$

Explanation: For the correct equation, multiplying by

(x-2)(x+3)

gives

x(x+3)-4(x-2)=10

, so

x^2-x-2=0

, with roots

2

and

-1

. The value

2

makes

x-2

zero, so it is rejected; the valid solution is

-1

. The other equations produce different roots, or produce

-3

and

2

, both of which are excluded.