SAT Math: Equivalent expressions

Equivalent expressions: expanding, factoring and rational expressions.

Advanced Math · Equivalent expressions
Both modules
15 practice questions · 4 with free solutions
Calculator allowed (Desmos)

Practise equivalent expressions →Adaptive SAT module

What the test covers

Expanding, factoring, rational expressions, exponent rules
Matching coefficients of identities

Key ideas

$(a \pm b)^{2} = a^{2} \pm 2 a b + b^{2}$ and $a^{2} - b^{2} = (a - b) (a + b)$ .
Two polynomials are equal for all $x$ exactly when their coefficients match, term by term.
Simplify a rational expression by factorising the top and bottom and cancelling common factors (for the allowed values of $x$ ).
Exponent rules: $x^{a} \cdot x^{b} = x^{a + b}$ , $\frac{x^{a}}{x^{b}} = x^{a - b}$ , $x^{1 / n} = \sqrt[n]{x}$ .

Common mistakes

$(3 x - 2)^{2} \neq 9 x^{2} + 4$ : don't forget the middle term.
Subtracting a product: bracket it first, then change every sign.

Do it in Desmos

Graph the original expression and each option; the equivalent one draws exactly the same curve.

Worked example

Which expression is equivalent to $4 x^{2} - 25$ ?

$(4 x - 5) (x + 5)$
$(4 x + 5) (x - 5)$
$(2 x - 5)^{2}$
$(2 x - 5) (2 x + 5)$

Answer: D: $(2 x - 5) (2 x + 5)$

Difference of two squares: $(2 x)^{2} - 5^{2} = (2 x - 5) (2 x + 5)$ .

Practice questions

Try each question before opening the solution. Desmos or your own calculator is allowed on every SAT Math question. Solutions to the first 3 are open to everyone; the rest need only a free account. Every answer here was checked independently before it was published.

Question 1

Which expression is equivalent to $(2 x - 5)^{2} - (x + 4) (x - 4)$ ?

$3 x^{2} - 20 x + 9$
$3 x^{2} - 28 x + 9$
$3 x^{2} + 41$
$3 x^{2} - 20 x + 41$

Show the answer and solution

Answer: D: $3 x^{2} - 20 x + 41$

$(2 x - 5)^{2} = 4 x^{2} - 20 x + 25$ and $(x + 4) (x - 4) = x^{2} - 16$ . The difference is $3 x^{2} - 20 x + 41$ .

Question 2

For all values of $x$ , $(2 x + a) (x - 3) = 2 x^{2} + b x - 15$ , where $a$ and $b$ are constants. What is the value of $b$ ?

Student-produced response: type your answer.

Show the answer and solution

Answer: -1

Expanding: $2 x^{2} + (a - 6) x - 3 a$ . Matching constants, $- 3 a = - 15$ , so $a = 5$ ; then $b = a - 6 = - 1$ .

Question 3

For all values of $x$ , $(4 x + a) (x - 2) = 4 x^{2} + b x - 2$ , where $a$ and $b$ are constants. What is the value of $b$ ?

Student-produced response: type your answer.

Show the answer and solution

Answer: -7

Expanding: $4 x^{2} + (a - 8) x - 2 a$ . Matching constants, $- 2 a = - 2$ , so $a = 1$ ; then $b = a - 8 = - 7$ .