SAT Math skill page

SAT Equivalent Expressions Practice

Recognize when two algebra expressions say the same thing in different forms.

Practice equivalent expressions in Dolphin SAT Practice questions

10-16 min practice time3 questions on pageAdvanced Math

Practice time10-16 min

Question bank3 questions

Best forAdvanced Math

What this tests

What to know for this SAT skill

Distribute and combine like terms
Factor common factors from expressions
Rewrite expressions to reveal a value
Check equivalence by expanding or substituting

Dolphin question bank

Practice your skill

Question 1 of 3Medium

The expression above is equivalent to ax^2 + bx + c, where a, b, and c are constants. What is the value of b?

1
2
$\frac{5}{2}$
3

Show answer and solution

Correct answer $C . \frac{5}{2}$

Solution walkthrough

The correct answer is $\frac{5}{2}$ . The expression $(\frac{1}{2} x + \frac{3}{2}) (\frac{3}{2} x + \frac{1}{2})$ can be written in the form $a x^{2} + b x + c$ , where $a$ , $b$ , and $c$ are constants, by multiplying out the expression using the distributive property of multiplication over addition.
The result is $(\frac{1}{2} x) (\frac{3}{2} x) + (\frac{1}{2} x) (\frac{1}{2}) + (\frac{3}{2}) (\frac{3}{2} x) + (\frac{3}{2}) (\frac{1}{2})$ . This expression can be rewritten by multiplying as indicated to give $\frac{3}{4} x^{2} + \frac{1}{4} x + \frac{9}{4} x + \frac{3}{4}$ , which can be simplified to $\frac{3}{4} x^{2} + \frac{10}{4} x + \frac{3}{4}$ , or $\frac{3}{4} x^{2} + \frac{5}{2} x + \frac{3}{4}$ .
This is in the form $a x^{2} + b x + c$ , where the value of $b$ is $\frac{5}{2}$ . Note that 5/2 and 2.5 are examples of ways to enter a correct answer.

Question 2 of 3Hard

\frac{x ^{2} - c}{x - b}

In the expression above, b and c are positive integers. If the expression is equivalent to x + b and

x \neq = b

, which of the following could be the value of c?

Show answer and solution

Correct answerA. 4

Solution walkthrough

If the given expression is equivalent to $x + b$ , then $\frac{x ^{2} - c}{x - b} = x + b$ , where $x$ isn't equal to $b$ .
Multiplying both sides of this equation by $x - b$ yields $x^{2} - c = (x + b) (x - b)$ .
Since the right-hand side of this equation is in factored form for the difference of squares, the value of $c$ must be a perfect square.
Only choice A gives a perfect square for the value of $c$ .

Why the other choices miss

Choices B, C, and D are incorrect. None of these values of

c

produces a difference of squares. For example, when 6 is substituted for

c

in the given expression, the result is

\frac{x ^{2} - 6}{x - b}

. The expression

x^{2} - 6

can't be factored with integer values, and therefore

\frac{x ^{2} - 6}{x - b}

isn't equivalent to

x + b

Question 3 of 3Hard

The expression above can be written in the form ay^2 + b, where a and b are constants. What is the value of a + b?

6532
6632
6732
6832

Show answer and solution

Correct answerB. 6632

Solution walkthrough

The correct answer is 6632.
Applying the distributive property to the expression yields $(7532 + 100 y^{2}) + (100 y^{2} - 1100)$ .
Then adding together $7532 + 100 y^{2}$ and $100 y^{2} - 1100$ and collecting like terms results in $200 y^{2} + 6432$ .
This is written in the form $a y^{2} + b$ , where $a = 200$ and $b = 6432$ .
Therefore $a + b = 200 + 6432 = 6632$ .

Avoid these traps

Common mistakes on this skill

Checking only one input value

Two expressions are equivalent only if they have the same value for every allowed input, not just one convenient test value.

Losing a negative sign while distributing

A factor outside parentheses multiplies every term inside, including each term following a minus sign.

Study plan

How to practice this skill in Dolphin

Choose whether distribution, factoring, or combining like terms best exposes the structure.
Rewrite one expression one operation at a time without changing its value.
Verify the final form with a second method or more than one permitted input value.

Practice equivalent expressions in Dolphin SAT

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FAQ

Questions about SAT Equivalent Expressions Practice

What makes two expressions equivalent?

They produce the same value for every input in their shared domain.

Can I prove equivalence by testing numbers?

Testing can catch an error, but algebraic rewriting is the reliable proof because one matching input is not sufficient.