PSAT Math Quiz: Equivalent Expressions

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Equivalent ExpressionsQuestion 1 of 20

Simplify the expression $\dfrac{6x^2y}{3xy^2}$ , assuming $x\ne 0$ and $y\ne 0$ . Which expression is equivalent?

\dfrac{2x}{y}

\dfrac{2y}{x}

\dfrac{3x}{2y}

\dfrac{2x^2}{y^2}

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PSAT Math Quiz

PSAT Math Quiz: Equivalent Expressions

Practice Equivalent Expressions in PSAT Math 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 Equivalent Expressions, giving you a quick way to practice the rules, question types, and explanations that matter most for PSAT Math.

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

Simplify the expression $\dfrac{6x^2y}{3xy^2}$ , assuming $x\ne 0$ and $y\ne 0$ . Which expression is equivalent?

$\dfrac{2x}{y}$ (correct answer)
$\dfrac{2y}{x}$
$\dfrac{3x}{2y}$
$\dfrac{2x^2}{y^2}$

Explanation: The question requires simplifying (

\frac{6x^2 y}{3x y^2}

) to find the equivalent expression, assuming x and y nonzero. Simplify coefficients: (

\frac{6}{3}

= 2); for x terms, (

\frac{x^2}{x}

= x); for y terms, (

\frac{y}{y^2}

\frac{1}{y}

). Combine: (2 cdot x cdot

\frac{1}{y}

\frac{2x}{y}

). A key error is inverting the fraction, leading to (

\frac{2y}{x}

) or (

\frac{3x}{2y}

). Another mistake is squaring terms incorrectly to get (

\frac{2x^2}{y^2}

). Cancel common factors carefully before combining, and check by multiplying back.

Question 2

If $4(x-2)+k$ is equivalent to $4x+5$ for all values of $x$ , what is the value of $k$ ? (Choose the constant that makes the two expressions identical.)

$-3$
$3$
$5$
$13$ (correct answer)

Explanation: We need to find

k

such that

4(x-2)+k = 4x+5

for all values of

x

. First, expand the left side:

4(x-2)+k = 4x-8+k

. For this to equal

4x+5

, we need

4x-8+k = 4x+5

. The

4x

terms already match on both sides, so we need the constant terms to be equal:

-8+k = 5

. Solving for

k

k = 5+8 = 13

. A common error is forgetting to distribute the 4 to the -2, which would give

4x-2+k

instead of

4x-8+k

. Always verify by substituting back:

4(x-2)+13 = 4x-8+13 = 4x+5

✓.

Question 3

Which expression is equivalent to $9a^2-25$ ? Choose the fully factored form over the integers.

$(9a-25)(a+1)$
$(3a-5)(3a+5)$ (correct answer)
$(9a-5)(a+5)$
$(3a-5)^2$

Explanation: We need to factor

9a^2-25

, which is a difference of squares. The pattern for difference of squares is

A^2-B^2 = (A-B)(A+B)

. Here,

9a^2 = (3a)^2

and

25 = 5^2

, so we have

(3a)^2-5^2

. Applying the formula:

(3a)^2-5^2 = (3a-5)(3a+5)

. The fully factored form is

(3a-5)(3a+5)

. A common mistake is not recognizing that

9a^2 = (3a)^2

and trying to factor as

(9a-5)(a+5)

, which doesn't work. Always verify by expanding:

(3a-5)(3a+5) = 9a^2+15a-15a-25 = 9a^2-25

✓.

Question 4

For all $x\neq -3$ , which expression is equivalent to $\dfrac{x^{2}-9}{x+3}$ ?

$x-3$ (correct answer)
$x+3$
$x^{2}+3$
$x^{2}-3$

Explanation: This question tests your ability to simplify rational expressions by factoring and canceling common factors. When you see a fraction with polynomials, look for opportunities to factor the numerator or denominator. The key insight is recognizing that the numerator

x^2 - 9

is a difference of squares, which factors as

(x+3)(x-3)

. This gives us:

\frac{x^2-9}{x+3} = \frac{(x+3)(x-3)}{x+3}

Since we're told that

x \neq -3

, we know the denominator

(x+3)

is never zero, so we can safely cancel the common factor

(x+3)

from both numerator and denominator:

\frac{(x+3)(x-3)}{x+3} = x-3

Therefore, the answer is A)

x-3

. Looking at the wrong answers: B)

x+3

would result if you mistakenly canceled

(x-3)

instead of

(x+3)

, or confused which factor remains. C)

x^2+3

and D)

x^2-3

suggest you might have tried to "cancel" terms incorrectly without proper factoring—you cannot simply subtract or add constants to the original expression. Study tip: Always look for special factoring patterns like difference of squares (

a^2-b^2 = (a+b)(a-b)

) when simplifying rational expressions. Factor completely before canceling, and remember that you can only cancel common factors, not individual terms.

Question 5

The expression $x^{2}+10x+25-(x^{2}-6x+9)$ is equivalent to which of the following?

$16(x+1)$ (correct answer)
$4(x+1)$
$16(x-1)$
$4(x-1)$

Explanation: When you encounter an expression that needs to be simplified, especially one involving subtraction of grouped terms, your first step is to distribute the negative sign and combine like terms systematically. Let's work through

x^{2}+10x+25-(x^{2}-6x+9)

step by step. First, distribute the negative sign to each term in the second parentheses:

x^{2}+10x+25-x^{2}+6x-9

. Now combine like terms. The

x^2

terms cancel out:

x^{2}-x^{2}=0

. For the

x

terms:

10x+6x=16x

. For the constants:

25-9=16

. This gives us

16x+16

, which factors as

16(x+1)

. Looking at the answer choices, option A gives us

16(x+1)

, which matches our result perfectly. Option B,

4(x+1)

, has the correct form but the wrong coefficient—this might tempt you if you made an arithmetic error when combining the

x

terms or constants. Options C and D both have

(x-1)

instead of

(x+1)

, which would result from incorrectly handling the signs when distributing the negative or combining terms. The key strategy here is to work methodically: distribute any negative signs first, then group and combine like terms carefully. Double-check your arithmetic, especially when dealing with positive and negative terms. On the PSAT, many algebra errors come from rushing through the sign changes, so take your time with distribution and combination steps.

Question 6

Which of the following is equivalent to $\sqrt{50}-2\sqrt{8}+\sqrt{2}$ ?

$2\sqrt{2}$ (correct answer)
$\sqrt{2}$
$4\sqrt{2}$
$10\sqrt{2}$

Explanation: When you encounter radical expressions like this, you need to simplify each term by factoring out perfect squares from under the radicals, then combine like terms. Start by simplifying each radical separately. For

\sqrt{50}

, factor 50 as

25 \times 2

, so

\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}

. For

\sqrt{8}

, factor 8 as

4 \times 2

, so

\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}

. The term

\sqrt{2}

is already simplified. Now substitute these simplified forms:

5\sqrt{2} - 2(2\sqrt{2}) + \sqrt{2} = 5\sqrt{2} - 4\sqrt{2} + \sqrt{2}

. Since all terms have the same radical part (

\sqrt{2}

), you can combine the coefficients:

5 - 4 + 1 = 2

. Therefore, the expression equals

2\sqrt{2}

, which is choice A. Choice B (

\sqrt{2}

) would result from incorrectly combining coefficients as

5 - 4 - 1 = 0

and then adding 1, or from other arithmetic errors. Choice C (

4\sqrt{2}

) might come from forgetting the coefficient 2 in front of

\sqrt{8}

, making the calculation

5 - 2 + 1 = 4

. Choice D (

10\sqrt{2}

) could result from adding all coefficients without considering the subtraction:

5 + 4 + 1 = 10

. Study tip: Always factor out perfect squares first, then treat radical expressions like algebraic terms with the same variable—combine coefficients while keeping the radical part unchanged.

Question 7

If $2(x-3)+5x=7x+k$ for all values of $x$ , what is the value of the constant $k$ ? Distribute first, then compare the constant terms on both sides.

$-6$ (correct answer)
$-3$
$3$
$6$

Explanation: We need to find

k

such that

2(x-3)+5x=7x+k

for all values of

x

. First, distribute on the left side:

2(x-3) = 2x - 6

. The left side becomes

2x - 6 + 5x = 7x - 6

. For the equation

7x - 6 = 7x + k

to be true for all

x

, the coefficients of

x

must match (they do: both are 7) and the constant terms must match. Therefore,

-6 = k

, so

k = -6

. A common mistake is to write

k = 6

by forgetting the negative sign.

Question 8

The expression $(x-6)(x+2)$ is expanded and written in standard form $x^2+bx+c$ . What is the value of $b$ ?

$-12$
$-4$ (correct answer)
$4$
$12$

Explanation: We need to expand

(x-6)(x+2)

and identify the coefficient

b

in the standard form

x^2+bx+c

. Using FOIL: First terms give

x cdot x = x^2

, Outer terms give

x cdot 2 = 2x

, Inner terms give

-6 cdot x = -6x

, and Last terms give

-6 cdot 2 = -12

. Combining these:

x^2 + 2x - 6x - 12 = x^2 - 4x - 12

. The coefficient of

x

b = -4

. A common mistake is adding the inner and outer products incorrectly, getting

2x + 6x = 8x

instead of

2x - 6x = -4x

Question 9

The expression $x^2+12x+36$ can be rewritten in the form $(x+m)^2$ . What is the value of $m$ ?

$6$ (correct answer)
$12$
$18$
$-6$

Explanation: We need to find

m

such that

x^2+12x+36 = (x+m)^2

. Expanding the right side:

(x+m)^2 = x^2+2mx+m^2

. Comparing coefficients with

x^2+12x+36

, we need

2m = 12

(coefficient of

x

) and

m^2 = 36

(constant term). From

2m = 12

, we get

m = 6

. Let's verify: if

m = 6

, then

m^2 = 36

✓, confirming our answer. Therefore,

x^2+12x+36 = (x+6)^2

and

m = 6

. This is a perfect square trinomial, and recognizing the pattern

a^2+2ab+b^2 = (a+b)^2

helps identify

m

quickly.

Question 10

Simplify $\dfrac{3x^2-12x}{3x}$ , assuming $x\neq 0$ . Choose the expression that results after canceling common factors and simplifying completely.

$x-4$ (correct answer)
$x^2-4x$
$x-12$
$3x-12$

Explanation: The question asks to simplify (

dfrac{3x^2

- 12x}{3x}) for (x eq 0) by canceling common factors. First, factor the numerator: (3x^2 - 12x = 3x(x - 4)). Then the expression is (dfrac{3x(x - 4)}{3x}), and cancel 3x top and bottom, leaving (x - 4). A common error is canceling incorrectly, like dividing only one term, resulting in something like x - 12. Another mistake could be factoring out 3 instead of 3x, leading to x-12 or 3x-12. Always factor the numerator completely and cancel entire common factors. In tests, verify by plugging in a number like x=1 into original and simplified expressions.

Question 11

Which expression is equivalent to $2x(3x-1)-(x-2)(x+2)$ ? Fully expand each product, watch for the difference of squares in $(x-2)(x+2)$ , and then combine like terms.

$5x^2-2x-4$
$7x^2-2x+4$
$5x^2-2x+4$ (correct answer)
$7x^2+2x-4$

Explanation: To simplify

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

, first expand each product. For

2x(3x-1)

2x(3x) - 2x(1) = 6x^2 - 2x

. For

(x-2)(x+2)

, recognize this as a difference of squares:

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

. Now combine:

6x^2 - 2x - (x^2 - 4) = 6x^2 - 2x - x^2 + 4 = 5x^2 - 2x + 4

. The key insight is recognizing

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

has no middle term, and remembering to distribute the negative sign to get

+4

Question 12

Which expression is equivalent to $3(2x-5)-4(x+1)+2x$ ? Be careful to distribute the $3$ and the $-4$ correctly, and then combine like terms to write the result as a simplified linear expression in $x$ .

$4x-19$ (correct answer)
$4x-15$
$8x-19$
$4x+19$

Explanation: The task is to simplify

3(2x-5)-4(x+1)+2x

by distributing and combining like terms. First, distribute the 3:

3(2x-5) = 6x - 15

. Next, distribute the -4:

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

. Now combine all terms:

6x - 15 - 4x - 4 + 2x = (6x - 4x + 2x) + (-15 - 4) = 4x - 19

. A common error is distributing -4 as

-4x + 4

instead of

-4x - 4

, which would incorrectly yield

4x - 11

. When distributing a negative number, both terms inside the parentheses change sign.

Question 13

A student factors the expression $9y^2-25$ . Which expression is equivalent to $9y^2-25$ ?

$(9y-25)(y+1)$
$(3y-5)(3y+5)$ (correct answer)
$(9y-5)(y+5)$
$(3y-25)(3y+1)$

Explanation: The question requires factoring (

9y^2

- 25) to find the equivalent expression. Recognize this as a difference of squares: ( (3y)^2 - 5^2 ). Apply the formula (a^2 - b^2 = (a - b)(a + b)), so it factors to ((3y - 5)(3y + 5)). A key error is mismatching the factors, such as using 9y and 1 incorrectly to get ((9y - 25)(y + 1)). Another mistake is confusing it with sum of squares or incorrect coefficients like ((9y - 5)(y + 5)). To check, expand the factored form back to verify it matches the original.

Question 14

Expand and simplify the expression $(2x-3)(x+5)$ . Use FOIL carefully, paying attention to the sign of $-3$ and combining like terms into standard form $ax^2+bx+c$ .

$2x^2+7x-15$ (correct answer)
$2x^2-7x-15$
$2x^2+13x-15$
$2x^2+7x+15$

Explanation: To expand

(2x-3)(x+5)

, we use FOIL method systematically. First terms:

2x cdot x = 2x^2

. Outer terms:

2x cdot 5 = 10x

. Inner terms:

-3 cdot x = -3x

. Last terms:

-3 cdot 5 = -15

. Combining all terms:

2x^2 + 10x - 3x - 15 = 2x^2 + 7x - 15

. The most common error is getting the sign wrong on the last term, writing

+15

instead of

-15

because students forget that

(-3)(+5) = -15

Question 15

If $2(3x+k)-5x= x+8$ , what is the value of $k$ ? Distribute first, then combine like terms so both sides have matching coefficients of $x$ .

$1$
$2$
$4$ (correct answer)
$8$

Explanation: We need to solve

2(3x+k)-5x = x+8

for

k

. First, distribute on the left side:

6x + 2k - 5x = x + 8

. Combine like terms on the left:

(6x-5x) + 2k = x + 2k = x + 8

. Since the coefficients of

x

on both sides are already equal (both are 1), we need the constant terms to match:

2k = 8

. Dividing both sides by 2 gives

k = 4

. The key insight is recognizing that once the

x

terms match, the constant terms must also be equal.

Question 16

Which expression is equivalent to $\dfrac{6x^2-24x}{3x}$ for $x\ne 0$ ? Simplify completely and write the result in the form $ax+b$ .

$2x-8$ (correct answer)
$2x+8$
$2x^2-8x$
$\dfrac{2x-8}{x}$

Explanation: We need to simplify

\frac{6x^2-24x}{3x}

for

x \neq 0

. First, factor the numerator:

6x^2-24x = 6x(x-4)

. So we have

\frac{6x(x-4)}{3x}

. Since

x \neq 0

, we can cancel the common factor

x

\frac{6x(x-4)}{3x} = \frac{6(x-4)}{3}

. Now simplify the coefficient:

\frac{6}{3} = 2

, giving us

2(x-4) = 2x-8

. The simplified expression is

2x-8

. A common error is dividing each term separately without factoring first, which can lead to mistakes with the algebra. Always factor before simplifying rational expressions.

Question 17

The expression $x^2+10x+25$ is a perfect square trinomial. Which expression is equivalent to $x^2+10x+25$ written as a squared binomial?

$(x+5)^2$ (correct answer)
$(x-5)^2$
$(x+25)(x+1)$
$(x+5)(x+25)$

Explanation: We need to recognize that

x^2 + 10x + 25

is a perfect square trinomial and write it as a squared binomial. A perfect square trinomial has the form

a^2 + 2ab + b^2 = (a+b)^2

. Here, the first term is

x^2 = (x)^2

and the last term is

25 = 5^2

. The middle term should be

2 cdot x cdot 5 = 10x

, which matches. Therefore,

x^2 + 10x + 25 = (x+5)^2

. Students often confuse the sign and write

(x-5)^2

, but that expands to

x^2 - 10x + 25

Question 18

Which expression is equivalent to $(2m-1)^2$ after expanding?

$4m^2-1$
$4m^2-4m+1$ (correct answer)
$4m^2-2m+1$
$2m^2-4m+1$

Explanation: The question asks for the expanded equivalent of ((2m - 1)^2). Expand using the square formula: ((2m)^2 - 2 cdot 2m cdot 1 + (-1)^2 = 4m^2 - 4m + 1). Alternatively, FOIL: First (2m cdot 2m =

4m^2

), Outer (2m cdot (-1) = -2m), Inner (-1 cdot 2m = -2m), Last (-1 cdot -1 = 1), then combine (

4m^2

- 2m - 2m + 1 =

4m^2

- 4m + 1). A common error is forgetting the middle term, resulting in (

4m^2

- 1). Another mistake is halving coefficients incorrectly, like (

4m^2

- 2m + 1). Always expand fully and combine like terms when squaring binomials.

Question 19

The expression $9y^2-25$ can be rewritten using a special factoring pattern. Which option gives a factored form that is equivalent to $9y^2-25$ ?

$(9y-25)(y+1)$
$(3y-5)(3y+5)$ (correct answer)
$(9y-5)(y+5)$
$(3y-5)^2$

Explanation: The expression

9y^2-25

is a difference of squares since

9y^2 = (3y)^2

and

25 = 5^2

. The difference of squares pattern states that

a^2-b^2 = (a-b)(a+b)

. Here,

a = 3y

and

b = 5

, so

9y^2-25 = (3y)^2-5^2 = (3y-5)(3y+5)

. A common mistake is trying to factor out a common factor when none exists, or not recognizing that

9y^2

is a perfect square. When you see a binomial with subtraction and both terms are perfect squares, always check for the difference of squares pattern.

Question 20

Which expression is equivalent to $(2x+3)(x-4) - (x-4)$ ? Factor or expand as needed, but choose the simplest equivalent expression.

$(x-4)(2x+2)$ (correct answer)
$(x-4)(2x+4)$
$2x^2-5x-12$
$2x^2-9x-12$

Explanation: We need to simplify

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

. Notice that

(x-4)

is a common factor. We can rewrite this as

(2x+3)(x-4) - 1(x-4) = (x-4)[(2x+3)-1] = (x-4)(2x+3-1) = (x-4)(2x+2)

. The expression simplifies to

(x-4)(2x+2)

. Alternatively, we could expand everything first:

(2x+3)(x-4) = 2x^2-8x+3x-12 = 2x^2-5x-12

, then subtract

(x-4)

to get

2x^2-5x-12-x+4 = 2x^2-6x-8

, which factors as

2(x^2-3x-4) = 2(x-4)(x+1) = (x-4)(2x+2)

. The factored form

(x-4)(2x+2)

is simpler than the expanded form.